**Subcategory Pollutant**  **No. of Sources**  **Industryw ide**  **No. of Sources**  **with Data Determination** 

**of No. of Sources in**  **MA CT Floor**  **No. of**

**Sources in MA** **CT Floor (N)**  **No. Data**

**Points in MA** **CT Floor (n)**  **Av erage** 

**of Top Sources**  **UPL Unit of Measure**  **Data Distribution**  All Kilns PM 225 31 12% Sources Industrywide 1 3 0.0060 0.018 lb/ton lognormal Equivalent PM 225 31 12% Sources Industrywide 1 3 0.0020 0.0059 gr/dscf @ 7% O2 lognormal Large Kilns Mercury 122 19 12% Sources with Data 1 3 1.5E-05 2.0E-05 lb/ton normal Equivalent Mercury 122 19 12% Sources with Data 1 3 10 13 μg/dscm @ 7% O2 lognormal Equivalent Mercury 122 19 12% Sources with Data 1 3 0.00018 0.00024 lb/hr normal Small Kilns Mercury 103 2 12% Sources with Data 1 3 1.8E-04 2.0E-04 lb/ton lognormal Equivalent Mercury 103 2 12% Sources with Data 1 3 62 70 μg/dscm @ 7% O2 normal Equivalent Mercury 103 2 12% Sources with Data 1 3 0.0010 0.0011 lb/hr lognormalRuns 514 PM lb/ton 514 PM conc 568 Hg Large lb/ton568 Hg Large conc

1 2.99E-03 1.02E-03 1.37E-05 1.17E+01 2 9.53E-03 3.18E-03 1.50E-05 9.98E+00 3 5.61E-03 1.78E-03 1.56E-05 9.71E+00 Average 6.04E-03 2.00E-03 1.48E-05 1.05E+01 Standard Deviation 3.29E-03 1.09E-03 9.40E-07 1.06E+00

1 8.74E-11 8.93E-13 1.13E-24 2.18E+00 2 1.48E-10 1.98E-12 2.34E-27 4.99E-02 3 3.43E-14 2.10E-15 4.31E-25 3.04E-01 Testing Normality Raw Data Raw Data Raw Data Raw Data Sample Size 3.00 3.00 3.00 3.00 Kurtosis -2.00 -2.00 -2.00 -2.00 SE Kurtosis 2.83 2.83 2.83 2.83 Result Kurtosis normal normal normal normal Skewness 0.58 0.85 -1.00 1.61 SE Skewness 1.22 1.22 1.22 1.22 Result Skewness normal normal normal normal lb/ton 0.471984818 0.690883834 0.812869883 1.312133767 Lognormal Lognormal Normal Lognormal 1 2 3 4568 Hg Large lb/hr531 Hg Small lb/ton531 Hg Small conc 531 Hg Small lb/hr Example 9

1.63E-04 1.84E-04 6.03E+01 1.04E-03 1.78E-04 1.69E-04 6.21E+01 9.55E-04 1.85E-04 1.76E-04 6.32E+01 9.96E-04 1.75E-04 1.76E-04 6.19E+01 9.97E-04 #DIV/0! 1.12E-05 7.42E-06 1.48E+00 4.20E-05 #DIV/0! 2.24E-20 3.16E-21 6.01E+00 3.23E-18 #DIV/0! 4.66E-23 2.91E-21 1.60E-03 2.98E-18 #DIV/0! 8.59E-21 5.41E-28 3.48E+00 5.53E-25 #DIV/0! Raw Data Raw Data Raw Data Raw Data Raw Data 3.00 3.00 3.00 3.00 0.00 -2.00 -2.00 -2.00 -2.00 #DIV/0! 2.83 2.83 2.83 2.83 #DIV/0! normal normal normal normal #DIV/0! -1.00 0.09 -0.60 0.09 #DIV/0! 1.22 1.22 1.22 1.22 0.00 normal normal normal normal #DIV/0! 0.812869883 0.075456359 0.488415014 0.075456359 #DIV/0! Normal Lognormal Normal Lognormal #DIV/0! 5 6 7 8 9Example 10 Example 11 Example 12 Example 13 Example 14

#DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! Raw Data Raw Data Raw Data Raw Data Raw Data 0.00 0.00 0.00 0.00 0.00 #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! 0.00 0.00 0.00 0.00 0.00 #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! 10 11 12 13 14Example 15 Example 16 Example 17 Example 18 Example 19

#DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! Raw Data Raw Data Raw Data Raw Data Raw Data 0.00 0.00 0.00 0.00 0.00 #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! 0.00 0.00 0.00 0.00 0.00 #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! 15 16 17 18 19Example 20 Example 21 Example 1 Example 2 -5.81E+00 -6.88E+00 -4.65E+00 -5.75E+00 -5.18E+00 -6.33E+00 #DIV/0! #DIV/0! -5.22E+00 -6.32E+00 #DIV/0! #DIV/0! 5.81E-01 5.67E-01 #DIV/0! #DIV/0! 1.27E-01 1.00E-01 #DIV/0! #DIV/0! 1.01E-01 1.06E-01 #DIV/0! #DIV/0! 1.31E-06 5.57E-09 Raw Data Raw Data Log Data Log Data 0.00 0.00 3.00 3.00 #DIV/0! #DIV/0! -2.00 -2.00 #DIV/0! #DIV/0! 2.83 2.83 #DIV/0! #DIV/0! normal normal #DIV/0! #DIV/0! -0.26 0.07 0.00 0.00 1.22 1.22 #DIV/0! #DIV/0! normal normal #DIV/0! #DIV/0! 0.213309407 0.055985172 #DIV/0! #DIV/0! 20 21Example 3 Example 4 Example 5 Example 6 Example 7

-1.12E+01 2.46E+00 -8.72E+00 -8.60E+00 4.10E+00 -1.11E+01 2.30E+00 -8.63E+00 -8.69E+00 4.13E+00 -1.11E+01 2.27E+00 -8.59E+00 -8.64E+00 4.15E+00 -1.11E+01 2.34E+00 -8.65E+00 -8.64E+00 4.12E+00 6.44E-02 9.91E-02 6.44E-02 4.21E-02 2.39E-02 2.53E-05 1.65E-04 2.53E-05 3.19E-06 4.19E-07 6.81E-08 3.40E-06 6.81E-08 3.11E-06 1.36E-10 9.00E-06 2.45E-05 9.00E-06 5.67E-15 2.35E-07 Log Data Log Data Log Data Log Data Log Data 3.00 3.00 3.00 3.00 3.00 -2.00 -2.00 -2.00 -2.00 -2.00 2.83 2.83 2.83 2.83 2.83 normal normal normal normal normal -1.06 1.58 -1.06 0.03 -0.63 1.22 1.22 1.22 1.22 1.22 normal normal normal normal normal 0.863705636 1.293580098 0.863705636 0.023935519 0.513997038Example 8 Example 9 Example 10 Example 11 Example 12

-6.87E+00 #NUM! #NUM! #NUM! #NUM! -6.95E+00 #NUM! #NUM! #NUM! #NUM! -6.91E+00 #NUM! #NUM! #NUM! #NUM! -6.91E+00 #NUM! #NUM! #NUM! #NUM! 4.21E-02 #NUM! #NUM! #NUM! #NUM! 3.19E-06 #NUM! #NUM! #NUM! #NUM! 3.11E-06 #NUM! #NUM! #NUM! #NUM! 5.67E-15 #NUM! #NUM! #NUM! #NUM! Log Data Log Data Log Data Log Data Log Data 3.00 0.00 0.00 0.00 0.00 -2.00 #NUM! #NUM! #NUM! #NUM! 2.83 #DIV/0! #DIV/0! #DIV/0! #DIV/0! normal #NUM! #NUM! #NUM! #NUM! 0.03 #NUM! #NUM! #NUM! #NUM! 1.22 0.00 0.00 0.00 0.00 normal #NUM! #NUM! #NUM! #NUM! 0.023935519 #NUM! #NUM! #NUM! #NUM!Example 13 Example 14 Example 15 Example 16 Example 17

#NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! Log Data Log Data Log Data Log Data Log Data 0.00 0.00 0.00 0.00 0.00 #NUM! #NUM! #NUM! #NUM! #NUM! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! 0.00 0.00 0.00 0.00 0.00 #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM!Example 18 Example 19 Example 20 Example 21

#NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! Log Data Log Data Log Data Log Data 0.00 0.00 0.00 0.00 #NUM! #NUM! #NUM! #NUM! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! 0.00 0.00 0.00 0.00 #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM!Raw Data lb/ton Runs 514 1 2.99E-03 2 9.53E-03 3 5.61E-03 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 Testing Normality Raw Data Logtransformed Data Sample Size 3.00 3.00 Kurtosis #DIV/0! #DIV/0! SE Kurtosis #DIV/0! #DIV/0! Average of Raw Data 6.04E-03 Result Kurtosis #DIV/0! #DIV/0! Skewness 0.58 -0.26 SE Skewness 1.22 1.22 Result Raw Data Result Log Data Result Skewness normal normal #DIV/0! #DIV/0! 1. Take LN( ) of Raw Data Runs 514 1 -5.81E+00 2 -4.65E+00 3 -5.18E+00 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 3

3. Calculate -5.22E+00

0.337445542 4. Calculate

5. m= number future runs = 3 6. Calculate Use for small sample sets that are Lognormal.

2. Calculate sample size, n=

1 ˆ n i

i x n     2 2 1 ˆ 1

n i i x x n   


 

### 

222 ˆˆˆ 4 3 2 233 1 3 1 e e e          

0900006481934216 PM-lbton 12/1/201412.2520905
0.483277475

27.35208269
7. Calculate 1.24411698

9. substitue the values of in cell D7 and F7 respectively

3.484z value from "z-stat"

a- Calculate 0.006424721
b- Calculate 4.1277E-05

c- Calculate 0.401363291

d-calculate 0.140949221

e- Calculate 0.010103

f- Calculate UPL 1.82E-02 11. In column B, go down the the row number from Step 10. above, and copy the z value. 12. calculate the UPL using the formula a- Numerator first term
b- denominator first term

8. Go to tab distribution for 99 percentile "z-stat"

10. In column AE indentify the value that is the smallest value that is larger than 0.99, note what row number this is. 222 ˆˆˆ 4 3 2 233 e e e   

###   2 2 ˆ 1 m e



2z    2 2 ˆ ˆ 1 1 2
z e e

m      

2z

 1z 

2 ˆ 2 2 2 2 2 4
ˆ ˆ ˆ ˆ ˆ ˆ 2 2 2 .99 ˆ ˆ

( 1) 2( 1) z UPL e me e m e

m n n                    

  

2 ˆ 2 ˆ e  
2 ˆ ˆ 2 e  

2 ˆ ( 1) e



2 4 ˆ ˆ 2( 1) n n       

  

2 2 2 2 4
ˆ ˆ ˆ ˆ ˆ 2 2 2 ˆ ˆ ( 1) 2( 1) me e me n n             

 

     2 2 2 ˆ 233 1 3 1 1 z e e e m

m e           



0900006481934216 PM-lbton 12/1/2014∆z z

0.101 27.35208 1.244117

-5=f(a) 463.2035 1 -4.899 421.0576 2 -4.798 381.7066 3 -4.697 345.03 4 -4.596 310.91 5 -4.495 279.2314 6 -4.394 249.8814 7 -4.293 222.7498 8 -4.192 197.7287 9 -4.091 174.7132 10 -3.99 153.6005 11 -3.889 134.2906 12 -3.788 116.6859 13 -3.687 100.6914 14 -3.586 86.21459 15 -3.485 73.16549 16 -3.384 61.4567 17 -3.283 51.00332 18 -3.182 41.723 19 -3.081 33.53592 20 -2.98 26.3648 21 -2.879 20.13487 22 -2.778 14.77392 23 -2.677 10.21226 24 -2.576 6.382752 25 -2.475 3.220766 26 -2.374 0.664222 27 -2.273 -1.34643 28 -2.172 -2.8682 29 -2.071 -3.95557 30 -1.97 -4.66049 31 -1.869 -5.03237 32 -1.768 -5.11808 33 -1.667 -4.96198 34 -1.566 -4.60588 35 -1.465 -4.08904 36 -1.364 -3.44822 37 -1.263 -2.71761 38 -1.162 -1.9289 39 -1.061 -1.11122 

#   1 3 

## 1 3 6 24 z 

## z z      

     2z  1z



## 40 -0.96 -0.29118

41 -0.859 0.507151 42 -0.758 1.262234 43 -0.657 1.955065 44 -0.556 2.569176 45 -0.455 3.090631 46 -0.354 3.508029 47 -0.253 3.812502 48 -0.152 3.997719 49 -0.051 4.05988 50 0.05 3.99772 51 0.151 3.812508 52 0.252 3.508048 53 0.353 3.090678 54 0.454 2.569268 55 0.555 1.955224 56 0.656 1.262487 57 0.757 0.50753 58 0.858 -0.29064 59 0.959 -1.11048 60 1.06 -1.92791 61 1.161 -2.71633 62 1.262 -3.44659 63 1.363 -4.08701 64 1.464 -4.60338 65 1.565 -4.95895 66 1.666 -5.11444 67 1.767 -5.02804 68 1.868 -4.6554 69 1.969 -3.94963 70 2.07 -2.86133 71 2.171 -1.33853 72 2.272 0.673248 73 2.373 3.23102 74 2.474 6.394343 75 2.575 10.2253 76 2.676 14.78852 77 2.777 20.15115 78 2.878 26.38289 79 2.979 33.55595 80 3.08 41.7451 81 3.181 51.02763 82 3.282 61.48336 83 3.383 73.19465 84 3.484 86.2464 85 3.585 100.726 86 3.686 116.7235 87 3.787 134.3314 88 3.888 153.644689 3.989 174.7607

90 4.09 197.7799 91 4.191 222.8047 92 4.292 249.9404 93 4.393 279.2947 94 4.494 310.9777 95 4.595 345.1022 96 4.696 381.7836 97 4.797 421.1397 98 4.898 463.2908 99 4.999 508.3599distribution of Z Normal distribution

1.48672E-06 0.000688654 2.45094E-06 0.001031987 3.9995E-06 0.001526636 6.46025E-06 0.002228979 1.03291E-05 0.003211417 1.63472E-05 0.004564666 2.56093E-05 0.006399279 3.97118E-05 0.008845786 6.09553E-05 0.012052605 9.26132E-05 0.016180747 0.000139285 0.021394248 0.000207351 0.027845257 0.000305546 0.035652911 0.000445674 0.04487554 0.000643469 0.055476427 0.000919619 0.067284381 0.001300942 0.079951609 0.001821704 0.092912943 0.002525035 0.10535202 0.003464389 0.116181478 0.004704958 0.124045247 0.006324913 0.127351286 0.008416337 0.124342289 0.011085658 0.113209674 0.014453386 0.092252383 0.018652949 0.06007678 0.023828414 0.015827368 0.030130931 -0.040569091 0.03771375 -0.108170447 0.046725789 -0.184827021 0.057303789 -0.267063539 0.069563239 -0.350067665 0.083588399 -0.427812432 0.099421884 -0.493329872 0.117054396 -0.539138381 0.136415346 -0.557808155 0.157365126 -0.542629187 0.179689839 -0.488327241 0.203099244 -0.391758021 0.227228529 -0.252500893 

# 

#  

2 4 2 

## 336 6 24 z 

## z z   

    

     

#   2 1 3 

### ( ) 1 3 ( )

6 24 z G 

### f z z z z             

## ( ) z 0.251644341 -0.073273625 0.275855243 0.139900292 0.299326439 0.377819919 0.321498297 0.628550081 0.341807853 0.878164461 0.359712192 1.11173754 0.37471238 1.314501734 0.386376486 1.473061259 0.394360216 1.576541345 0.398423793 1.617552685 0.398443914 1.592867053 0.394419966 1.503729262 0.386474058 1.355769593 0.374844865 1.158524603 0.359875719 0.924617077 0.34199778 0.668682393 0.32170943 0.406154008 0.299553264 0.152032198 0.276092166 -0.080243366 0.25188591 -0.279714154 0.227469632 -0.438541779 0.203335281 -0.552326028 0.17991684 -0.620099454 0.157579839 -0.64403015 0.136615273 -0.628891587 0.117237788 -0.581376068 0.099587708 -0.509335428 0.083736272 -0.421029404 0.069693339 -0.32445035 0.05741676 -0.226775201 0.046822635 -0.133974896 0.037795734 -0.050590672 0.030199481 0.02033173 0.023885038 0.077173047 0.018699163 0.119568861 0.014490659 0.148171378 0.011115369 0.164379871 0.008439746 0.170070626 0.006343145 0.167350506 0.004718997 0.158350443 0.003475077 0.145067465 0.002533081 0.129257107 0.001827693 0.112372732 0.001305351 0.09554474 0.000922829 0.079590688 0.00064578 0.065046903 0.00044732 0.052212756 0.000306705 0.041200145 0.000208159 0.0319824270.000139842 0.024438844 9.29928E-05 0.018392104 6.12113E-05 0.013638163 3.98826E-05 0.009968273 2.5722E-05 0.007184023 1.64209E-05 0.005106528 1.03767E-05 0.00358101 6.49066E-06 0.002478027 4.01874E-06 0.001692449 2.46298E-06 0.001141074 1.49417E-06 0.000759577Absolute values

0.000688654 3.4777E-05 0.001031987 0.000104231 0.001526636 0.00015419 0.002228979 0.000225127 0.003211417 0.000324353 0.004564666 0.000461031 0.006399279 0.000646327 0.008845786 0.000893424 0.012052605 0.001217313 0.016180747 0.001634255 0.021394248 0.002160819 0.027845257 0.002812371 0.035652911 0.003600944 0.04487554 0.004532429 0.055476427 0.005603119 0.067284381 0.006795723 0.079951609 0.008075112 0.092912943 0.009384207 0.10535202 0.010640554 0.116181478 0.011734329 0.124045247 0.01252857 0.127351286 0.01286248 0.124342289 0.012558571 0.113209674 0.011434177 0.092252383 0.009317491 0.06007678 0.006067755 0.015827368 0.001598564 0.040569091 0.004097478 0.108170447 0.010925215 0.184827021 0.018667529 0.267063539 0.026973417 0.350067665 0.035356834 0.427812432 0.043209056 0.493329872 0.049826317 0.539138381 0.054452976 0.557808155 0.056338624 0.542629187 0.054805548 0.488327241 0.049321051 0.391758021 0.03956756 0.252500893 0.02550259 

## 

#  

2 4 2 

### 336 ( ) 1 3 ( )

6 24 z 

### z z f z z z z    

       1 

## ( ) 2 () () 2 M i

i 

## xf a f x f b

 

##  

      

# 0.073273625 0.007400636 0.139900292 0.014129929 0.377819919 0.038159812 0.628550081 0.063483558 0.878164461 0.088694611 1.11173754 0.112285492 1.314501734 0.132764675 1.473061259 0.148779187 1.576541345 0.159230676 1.617552685 0.163372821 1.592867053 0.160879572 1.503729262 0.151876655 1.355769593 0.136932729 1.158524603 0.117010985 0.924617077 0.093386325 0.668682393 0.067536922 0.406154008 0.041021555 0.152032198 0.015355252 0.080243366 0.00810458 0.279714154 0.02825113 0.438541779 0.04429272 0.552326028 0.055784929 0.620099454 0.062630045 0.64403015 0.065047045 0.628891587 0.06351805 0.581376068 0.058718983 0.509335428 0.051442878 0.421029404 0.04252397 0.32445035 0.032769485 0.226775201 0.022904295 0.133974896 0.013531464 0.050590672 0.005109658 0.02033173 0.002053505 0.077173047 0.007794478 0.119568861 0.012076455 0.148171378 0.014965309 0.164379871 0.016602367 0.170070626 0.017177133 0.167350506 0.016902401 0.158350443 0.015993395 0.145067465 0.014651814 0.129257107 0.013054968 0.112372732 0.011349646 0.09554474 0.009650019 0.079590688 0.00803866 0.065046903 0.006569737 0.052212756 0.005273488 0.041200145 0.004161215 0.031982427 0.0032302250.024438844 0.002468323 0.018392104 0.001857603 0.013638163 0.001377455 0.009968273 0.001006796 0.007184023 0.000725586 0.005106528 0.000515759 0.00358101 0.000361682 0.002478027 0.000250281 0.001692449 0.000170937 0.001141074 0.000115248 0.000759577 3.83586E-05Cumulative normalization
3.4777E-05 1.13583E-05 0.000139008 4.54005E-05 0.000293198 9.57597E-05 0.000518325 0.000169287 0.000842678 0.000275222 0.001303709 0.000425797 0.001950036 0.00063689 0.002843461 0.000928686 0.004060774 0.001326266 0.005695029 0.001860021 0.007855848 0.002565753 0.010668219 0.003484285 0.014269163 0.004660369 0.018801593 0.006140679 0.024404712 0.007970682 0.031200435 0.010190193 0.039275547 0.012827559 0.048659754 0.01589248 0.059300308 0.01936773 0.071034638 0.023200212 0.083563208 0.027292095 0.096425687 0.031493035 0.108984259 0.035594717 0.120418436 0.039329168 0.129735926 0.042372299 0.135803681 0.044354054 0.137402245 0.044876151 0.141499723 0.046214405 0.152424939 0.049782626 0.171092468 0.05587952 0.198065885 0.064689152 0.233422719 0.076236842 0.276631775 0.090349102 0.326458092 0.106622587 0.380911069 0.124407158 0.437249692 0.142807589 0.49205524 0.160707312 0.541376291 0.176815775 0.580943851 0.189738706 0.606446442 0.19806796 

## ( ) 2 () () f a f x f b        0.613847078 0.200485039 0.627977007 0.205099933 0.666136819 0.217563088 0.729620377 0.238297085 0.818314988 0.267265118 0.930600479 0.303938032 1.063365155 0.347299534 1.212144342 0.395891443 1.371375017 0.447896851 1.534747839 0.501255102 1.695627411 0.553799048 1.847504066 0.603402602 1.984436795 0.648125408 2.10144778 0.686341688 2.194834105 0.716842054 2.262371027 0.738899897 2.303392582 0.75229771 2.318747834 0.7573128 2.326852414 0.759959789 2.355103543 0.769186727 2.399396263 0.783652916 2.455181192 0.801872508 2.517811236 0.822327745 2.582858282 0.843572383 2.646376332 0.864317646 2.705095315 0.883495513 2.756538193 0.900296973 2.799062163 0.914185481 2.831831648 0.924888132 2.854735943 0.932368772 2.868267408 0.936788205 2.873377066 0.938457041 2.87543057 0.939127724 2.883225048 0.941673433 2.895301503 0.945617654 2.910266812 0.950505387 2.926869179 0.95592779 2.944046312 0.961537914 2.960948714 0.967058309 2.976942108 0.972281819 2.991593922 0.977067163 3.00464889 0.981330971 3.015998536 0.985037813 3.025648555 0.98818955 3.033687214 0.99081501 3.040256952 0.992960714 3.04553044 0.994683057 3.049691655 0.996042127 3.05292188 0.9970971323.055390203 0.997903297 3.057247806 0.998509998 3.05862526 0.99895988 3.059632056 0.999288704 3.060357642 0.999525684 3.060873401 0.999694133 3.061235083 0.99981226 3.061485364 0.999894002 3.061656301 0.999949831 3.06177155 0.999987472 3.061809909 1 153.2361334Raw Data ranked by lb/ton, use concentration data Runs 514 1 1.02E-03 2 3.18E-03 3 1.78E-03 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 Testing Normality Raw Data Logtransformed Data Sample Size 3.00 3.00 Kurtosis #DIV/0! #DIV/0! SE Kurtosis #DIV/0! #DIV/0! Average of Raw Data 2.00E-03 Result Kurtosis #DIV/0! #DIV/0! Skewness 0.85 0.07 SE Skewness 1.22 1.22 Result Raw Data Result Log Data Result Skewness normal normal #DIV/0! #DIV/0! 1. Take LN( ) of Raw Data Runs 514 1 -6.88E+00 2 -5.75E+00 3 -6.33E+00 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 3

3. Calculate -6.32E+00

0.321232234 4. Calculate

5. m= number future runs = 3 6. Calculate Use for small sample sets that are Lognormal.

2. Calculate sample size, n=

1 ˆ n i

i x n     2 2 1 ˆ 1

n i i x x n   


 

### 

222 ˆˆˆ 4 3 2 233 1 3 1 e e e          

0900006481934216 PM-conc 12/1/201411.56063048
0.430526856

28.85228652
7. Calculate 1.200673476

9. substitue the values of in cell D7 and F7 respectively

3.484z value from "z-stat"

a- Calculate 0.002111346
b- Calculate 4.45778E-06

c- Calculate 0.378825754

d-calculate 0.132874948

e- Calculate 0.003224

f- Calculate UPL 5.86E-03 11. In column B, go down the the row number from Step 10. above, and copy the z value. 12. calculate the UPL using the formula a- Numerator first term
b- denominator first term

8. Go to tab distribution for 99 percentile "z-stat"

10. In column AE indentify the value that is the smallest value that is larger than 0.99, note what row number this is. 222 ˆˆˆ 4 3 2 233 e e e   

###   2 2 ˆ 1 m e



2z    2 2 ˆ ˆ 1 1 2
z e e

m      

2z

 1z 

2 ˆ 2 2 2 2 2 4
ˆ ˆ ˆ ˆ ˆ ˆ 2 2 2 .99 ˆ ˆ

( 1) 2( 1) z UPL e me e m e

m n n                    

  

2 ˆ 2 ˆ e  
2 ˆ ˆ 2 e  

2 ˆ ( 1) e



2 4 ˆ ˆ 2( 1) n n       

  

2 2 2 2 4
ˆ ˆ ˆ ˆ ˆ 2 2 2 ˆ ˆ ( 1) 2( 1) me e me n n             

 

     2 2 2 ˆ 233 1 3 1 1 z e e e m

m e           



0900006481934216 PM-conc 12/1/2014∆z z

0.101 28.85229 1.200673

-5=f(a) 493.879 1 -4.899 448.9942 2 -4.798 407.0825 3 -4.697 368.0158 4 -4.596 331.6691 5 -4.495 297.9197 6 -4.394 266.6478 7 -4.293 237.7364 8 -4.192 211.0708 9 -4.091 186.5395 10 -3.99 164.0332 11 -3.889 143.4457 12 -3.788 124.6732 13 -3.687 107.6147 14 -3.586 92.17184 15 -3.485 78.24911 16 -3.384 65.75353 17 -3.283 54.59484 18 -3.182 44.6855 19 -3.081 35.94065 20 -2.98 28.27811 21 -2.879 21.61841 22 -2.778 15.88474 23 -2.677 11.00301 24 -2.576 6.901806 25 -2.475 3.512406 26 -2.374 0.768779 27 -2.273 -1.39242 28 -2.172 -3.03183 29 -2.071 -4.20742 30 -1.97 -4.97447 31 -1.869 -5.38554 32 -1.768 -5.49053 33 -1.667 -5.33665 34 -1.566 -4.96839 35 -1.465 -4.42759 36 -1.364 -3.75336 37 -1.263 -2.98215 38 -1.162 -2.1477 39 -1.061 -1.28108 

#   1 3 

## 1 3 6 24 z 

## z z      

     2z  1z



## 40 -0.96 -0.41066

41 -0.859 0.437898 42 -0.758 1.241592 43 -0.657 1.980129 44 -0.556 2.6359 45 -0.455 3.193989 46 -0.354 3.642168 47 -0.253 3.970899 48 -0.152 4.173336 49 -0.051 4.245323 50 0.05 4.185393 51 0.151 3.994769 52 0.252 3.677366 53 0.353 3.239788 54 0.454 2.691329 55 0.555 2.043973 56 0.656 1.312396 57 0.757 0.513962 58 0.858 -0.33127 59 0.959 -1.20057 60 1.06 -2.06848 61 1.161 -2.90689 62 1.262 -3.68497 63 1.363 -4.36923 64 1.464 -4.92347 65 1.565 -5.3088 66 1.666 -5.48364 67 1.767 -5.40372 68 1.868 -5.0221 69 1.969 -4.28911 70 2.07 -3.15244 71 2.171 -1.55704 72 2.272 0.554806 73 2.373 3.243498 74 2.474 6.572137 75 2.575 10.60651 76 2.676 15.4151 77 2.777 21.06908 78 2.878 27.64229 79 2.979 35.2113 80 3.08 43.85534 81 3.181 53.65635 82 3.282 64.69893 83 3.383 77.0704 84 3.484 90.86077 85 3.585 106.1627 86 3.686 123.0716 87 3.787 141.6856 88 3.888 162.105389 3.989 184.4343

90 4.09 208.7787 91 4.191 235.2473 92 4.292 263.9516 93 4.393 295.0059 94 4.494 328.527 95 4.595 364.6346 96 4.696 403.451 97 4.797 445.101 98 4.898 489.7124 99 4.999 537.4155distribution of Z Normal distribution

1.48672E-06 0.00073426 2.45094E-06 0.001100458 3.9995E-06 0.001628127 6.46025E-06 0.002377473 1.03291E-05 0.003425839 1.63472E-05 0.004870167 2.56093E-05 0.006828654 3.97118E-05 0.009440931 6.09553E-05 0.012865876 9.26132E-05 0.017276016 0.000139285 0.022847365 0.000207351 0.029743563 0.000305546 0.038093374 0.000445674 0.047961043 0.000643469 0.059309735 0.000919619 0.071959376 0.001300942 0.085541534 0.001821704 0.099455631 0.002525035 0.112832439 0.003464389 0.124512395 0.004704958 0.133047326 0.006324913 0.136734543 0.008416337 0.133691335 0.011085658 0.121975616 0.014453386 0.099754472 0.018652949 0.065516732 0.023828414 0.018318787 0.030130931 -0.041954788 0.03771375 -0.114341673 0.046725789 -0.196595176 0.057303789 -0.285055794 0.069563239 -0.374635636 0.083588399 -0.458944983 0.099421884 -0.530579646 0.117054396 -0.581572166 0.136415346 -0.603990604 0.157365126 -0.590647504 0.179689839 -0.535861365 0.203099244 -0.436196588 0.227228529 -0.291098433 

# 

#  

2 4 2 

## 336 6 24 z 

## z z   

    

     

#   2 1 3 

### ( ) 1 3 ( )

6 24 z G 

### f z z z z             

## ( ) z 0.251644341 -0.103339333 0.275855243 0.12079643 0.299326439 0.371641392 0.321498297 0.636608094 0.341807853 0.900971442 0.359712192 1.148916769 0.37471238 1.364765255 0.386376486 1.534261981 0.394360216 1.645797856 0.398423793 1.691437805 0.398443914 1.667644387 0.394419966 1.575616834 0.386474058 1.421206737 0.374844865 1.214417959 0.359875719 0.968543939 0.34199778 0.699034335 0.32170943 0.422210168 0.299553264 0.153958978 0.276092166 -0.091462079 0.25188591 -0.302405574 0.227469632 -0.470516011 0.203335281 -0.591072493 0.17991684 -0.662988673 0.157579839 -0.688502973 0.136615273 -0.672621059 0.117237788 -0.622391438 0.099587708 -0.546102677 0.083736272 -0.45248751 0.069693339 -0.35000673 0.05741676 -0.246267083 0.046822635 -0.147605411 0.037795734 -0.058849316 0.030199481 0.016754864 0.023885038 0.077471072 0.018699163 0.122893462 0.014490659 0.153695357 0.011115369 0.171344555 0.008439746 0.177817661 0.006343145 0.175339083 0.004718997 0.166162026 0.003475077 0.152400712 0.002533081 0.135915856 0.001827693 0.118249807 0.001305351 0.100603957 0.000922829 0.083848961 0.00064578 0.068557805 0.00044732 0.055052392 0.000306705 0.043455725 0.000208159 0.0337436070.000139842 0.025791619 9.29928E-05 0.019414913 6.12113E-05 0.014399788 3.98826E-05 0.010527075 2.5722E-05 0.007588148 1.64209E-05 0.005394705 1.03767E-05 0.003783692 6.49066E-06 0.002618663 4.01874E-06 0.001788744 2.46298E-06 0.001206149 1.49417E-06 0.000802991Absolute values

0.00073426 3.70801E-05 0.001100458 0.000111146 0.001628127 0.000164441 0.002377473 0.000240125 0.003425839 0.00034601 0.004870167 0.000491887 0.006828654 0.000689694 0.009440931 0.000953534 0.012865876 0.001299454 0.017276016 0.001744878 0.022847365 0.002307584 0.029743563 0.0030041 0.038093374 0.003847431 0.047961043 0.004844065 0.059309735 0.005990283 0.071959376 0.007267897 0.085541534 0.008639695 0.099455631 0.010045019 0.112832439 0.011396076 0.124512395 0.012575752 0.133047326 0.01343778 0.136734543 0.013810189 0.133691335 0.013502825 0.121975616 0.012319537 0.099754472 0.010075202 0.065516732 0.00661719 0.018318787 0.001850198 0.041954788 0.004237434 0.114341673 0.011548509 0.196595176 0.019856113 0.285055794 0.028790635 0.374635636 0.037838199 0.458944983 0.046353443 0.530579646 0.053588544 0.581572166 0.058738789 0.603990604 0.061003051 0.590647504 0.059655398 0.535861365 0.054121998 0.436196588 0.044055855 0.291098433 0.029400942 

## 

#  

2 4 2 

### 336 ( ) 1 3 ( )

6 24 z 

### z z f z z z z    

       1 

## ( ) 2 () () 2 M i

i 

## xf a f x f b

 

##  

      

# 0.103339333 0.010437273 0.12079643 0.012200439 0.371641392 0.037535781 0.636608094 0.064297417 0.900971442 0.090998116 1.148916769 0.116040594 1.364765255 0.137841291 1.534261981 0.15496046 1.645797856 0.166225583 1.691437805 0.170835218 1.667644387 0.168432083 1.575616834 0.1591373 1.421206737 0.14354188 1.214417959 0.122656214 0.968543939 0.097822938 0.699034335 0.070602468 0.422210168 0.042643227 0.153958978 0.015549857 0.091462079 0.00923767 0.302405574 0.030542963 0.470516011 0.047522117 0.591072493 0.059698322 0.662988673 0.066961856 0.688502973 0.0695388 0.672621059 0.067934727 0.622391438 0.062861535 0.546102677 0.05515637 0.45248751 0.045701238 0.35000673 0.03535068 0.246267083 0.024872975 0.147605411 0.014908147 0.058849316 0.005943781 0.016754864 0.001692241 0.077471072 0.007824578 0.122893462 0.01241224 0.153695357 0.015523231 0.171344555 0.0173058 0.177817661 0.017959584 0.175339083 0.017709247 0.166162026 0.016782365 0.152400712 0.015392472 0.135915856 0.013727501 0.118249807 0.011943231 0.100603957 0.010161 0.083848961 0.008468745 0.068557805 0.006924338 0.055052392 0.005560292 0.043455725 0.004389028 0.033743607 0.0034081040.025791619 0.002604953 0.019414913 0.001960906 0.014399788 0.001454379 0.010527075 0.001063235 0.007588148 0.000766403 0.005394705 0.000544865 0.003783692 0.000382153 0.002618663 0.000264485 0.001788744 0.000180663 0.001206149 0.000121821 0.000802991 4.0551E-05Cumulative normalization
3.70801E-05 1.14751E-05 0.000148226 4.58713E-05 0.000312667 9.67604E-05 0.000552792 0.000171071 0.000898802 0.00027815 0.001390689 0.000430373 0.002080383 0.000643811 0.003033917 0.000938899 0.00433337 0.001341038 0.006078248 0.001881021 0.008385832 0.002595143 0.011389931 0.003524815 0.015237362 0.00471547 0.020081428 0.006214552 0.026071711 0.008068351 0.033339608 0.010317529 0.041979303 0.012991235 0.052024321 0.016099844 0.063420398 0.01962656 0.07599615 0.023518348 0.08943393 0.027676906 0.103244119 0.031950713 0.116746943 0.0361294 0.129066481 0.039941898 0.139141682 0.043059847 0.145758872 0.045107653 0.14760907 0.045680229 0.151846503 0.046991578 0.163395012 0.050565467 0.183251125 0.056710292 0.21204176 0.065620061 0.249879959 0.077329759 0.296233403 0.09167465 0.349821947 0.10825857 0.408560736 0.126436323 0.469563787 0.145314793 0.529219185 0.163776208 0.583341183 0.180525214 0.627397038 0.194159076 0.65679798 0.203257715 

## ( ) 2 () () f a f x f b        0.667235252 0.206487713 0.679435692 0.210263355 0.716971472 0.221879464 0.78126889 0.241777433 0.872267006 0.269938405 0.988307599 0.305849212 1.12614889 0.34850663 1.28110935 0.396461877 1.447334933 0.447903314 1.618170152 0.500771285 1.786602235 0.552895563 1.945739535 0.602143406 2.089281415 0.646564972 2.211937629 0.6845231 2.309760567 0.714796133 2.380363035 0.736645312 2.423006262 0.74984201 2.438556119 0.754654187 2.447793789 0.757512947 2.478336752 0.766965006 2.525858869 0.781671563 2.585557191 0.800146261 2.652519047 0.820868788 2.722057847 0.842388796 2.789992574 0.863412395 2.852854109 0.882866005 2.908010479 0.899935117 2.953711718 0.914078171 2.989062398 0.925018062 3.013935373 0.932715443 3.02884352 0.937329031 3.0347873 0.939168439 3.036479542 0.939692133 3.04430412 0.942113587 3.05671636 0.945954773 3.072239591 0.950758711 3.089545391 0.956114296 3.107504975 0.961672206 3.125214222 0.967152645 3.141996587 0.972346243 3.157389059 0.977109715 3.17111656 0.981357932 3.18305979 0.985053976 3.19322079 0.988198476 3.201689535 0.990819279 3.208613873 0.992962137 3.214174165 0.994682867 3.218563193 0.99604113 3.221971298 0.9970958283.224576251 0.997901976 3.226537157 0.998508813 3.227991536 0.998958896 3.229054771 0.999287933 3.229821174 0.99952511 3.230366039 0.999693728 3.230748192 0.999811992 3.231012676 0.999893842 3.23119334 0.999949751 3.231315161 0.999987451 3.231355712 1 161.833731Raw Data Runs 568 1 1.37E-05 2 1.50E-05 3 1.56E-05 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 Testing Normality Raw Data Sample Size 3.00 Kurtosis #DIV/0! SE Kurtosis #DIV/0! Result Kurtosis #DIV/0! Skewness -1.00 SE Skewness 1.22 Result Raw Data Result Skewness normal #DIV/0!
ni = number test runs = 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Total Number of sources = 1 n =Total # test runs = 3

2
This step is for QC only: means 1.48E-05 #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

1.48E-05 Mean =
Pooled Variance = 8.83E-13

m= number future runs = 3 Term1 6.67E-01
Term2 5.89E-13

Squared root Term2 7.67E-07 NOTE: the pvalue for the t-statistic is calculated as: 2*alpha, where 1-alpha is desired confidence, so if 99% confidence is desired then alpha=0.01 and 2*alpha=2*(0.01) t-statistic = quantile t-distribution with df degrees of freedom at .99 confidence level = 6.964556734 99 UPL POOLED VARIANCE = 2.01E-05 Use for large sample sets and small sample sets that are Normal.

 2 2 1 11n i i s x X n   

 1 1 1 i

n N ij i j X x n   

1 1 n m 

21 1 sn m        1 1 N i i df n

         

df t 2 1 1

s n m        1 1 i n i i

k k i x x n  1

N i in

  N

2

, 1 1 dfp X t s n m    

   

0900006481934216 Hg-Large-lbton 12/1/2014Raw Data Runs 568 1 1.17E+01 2 9.98E+00 3 9.71E+00 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 Testing Normality Raw Data Logtransformed Data Sample Size 3.00 3.00 Kurtosis #DIV/0! #DIV/0! SE Kurtosis #DIV/0! #DIV/0! Average of Raw Data 1.05E+01 Result Kurtosis #DIV/0! #DIV/0! variance 1.13 Skewness 1.61 1.58 SE Skewness 1.22 1.22 Result Raw Data Result Log Data Result Skewness normal normal #DIV/0! #DIV/0! 1. Take LN( ) of Raw Data Runs 568 1 2.46E+00 2 2.30E+00 3 2.27E+00 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 3

3. Calculate 2.34E+00

0.009813596 4. Calculate

5. m= number future runs = 3 6. Calculate Use for small sample sets that are Lognormal.

2. Calculate sample size, n=

1 ˆ n i

i x n     2 2 1 ˆ 1

n i i x x n   


 

### 

222 ˆˆˆ 4 3 2 233 1 3 1 e e e          

0900006481934216 Hg-Large-conc 12/1/20143.159255142
0.000291772

10829.83433
7. Calculate 0.172570439

9. substitue the values of in cell D7 and F7 respectively

3.484z value from "z-stat"

a- Calculate 10.46892065
b- Calculate 109.5982996

c- Calculate 0.009861907

d-calculate 0.003295275

e- Calculate 2.548128

f- Calculate UPL 1.34E+01 11. In column B, go down the the row number from Step 10. above, and copy the z value. 12. calculate the UPL using the formula a- Numerator first term
b- denominator first term

8. Go to tab distribution for 99 percentile "z-stat"

10. In column AE indentify the value that is the smallest value that is larger than 0.99, note what row number this is. 222 ˆˆˆ 4 3 2 233 e e e   

###   2 2 ˆ 1 m e



2z    2 2 ˆ ˆ 1 1 2
z e e

m      

2z

 1z 

2 ˆ 2 2 2 2 2 4
ˆ ˆ ˆ ˆ ˆ ˆ 2 2 2 .99 ˆ ˆ

( 1) 2( 1) z UPL e me e m e

m n n                    

  

2 ˆ 2 ˆ e  
2 ˆ ˆ 2 e  

2 ˆ ( 1) e



2 4 ˆ ˆ 2( 1) n n       

  

2 2 2 2 4
ˆ ˆ ˆ ˆ ˆ 2 2 2 ˆ ˆ ( 1) 2( 1) me e me n n             

 

     2 2 2 ˆ 233 1 3 1 1 z e e e m

m e           



0900006481934216 Hg-Large-conc 12/1/2014∆z z

0.101 10829.83 0.17257

-5=f(a) 215632.3 1 -4.899 196238.2 2 -4.798 178114.4 3 -4.697 161206.6 4 -4.596 145462.1 5 -4.495 130828.8 6 -4.394 117256.2 7 -4.293 104694.7 8 -4.192 93095.75 9 -4.091 82412.08 10 -3.99 72597.49 11 -3.889 63606.92 12 -3.788 55396.4 13 -3.687 47923.14 14 -3.586 41145.42 15 -3.485 35022.7 16 -3.384 29515.53 17 -3.283 24585.6 18 -3.182 20195.73 19 -3.081 16309.85 20 -2.98 12893.05 21 -2.879 9911.502 22 -2.778 7332.541 23 -2.677 5124.613 24 -2.576 3257.292 25 -2.475 1701.281 26 -2.374 428.4088 27 -2.273 -588.37 28 -2.172 -1374.97 29 -2.071 -1956.2 30 -1.97 -2355.7 31 -1.869 -2596.02 32 -1.768 -2698.57 33 -1.667 -2683.64 34 -1.566 -2570.39 35 -1.465 -2376.84 36 -1.364 -2119.9 37 -1.263 -1815.35 38 -1.162 -1477.85 39 -1.061 -1120.91 

#   1 3 

## 1 3 6 24 z 

## z z      

     2z  1z



## 40 -0.96 -756.935

41 -0.859 -397.199 42 -0.758 -51.8453 43 -0.657 270.1075 44 -0.556 560.7674 45 -0.455 813.3691 46 -0.354 1022.274 47 -0.253 1182.97 48 -0.152 1292.072 49 -0.051 1347.322 50 0.05 1347.586 51 0.151 1292.86 52 0.252 1184.265 53 0.353 1024.05 54 0.454 815.5871 55 0.555 563.3792 56 0.656 273.0538 57 0.757 -48.6347 58 0.858 -393.805 59 0.959 -753.45 60 1.06 -1117.43 61 1.161 -1474.5 62 1.262 -1812.25 63 1.363 -2117.18 64 1.464 -2374.65 65 1.565 -2568.89 66 1.666 -2682.99 67 1.767 -2698.96 68 1.868 -2597.62 69 1.969 -2358.72 70 2.07 -1960.85 71 2.171 -1381.48 72 2.272 -596.962 73 2.373 417.4893 74 2.474 1687.781 75 2.575 3240.945 76 2.676 5105.145 77 2.777 7309.666 78 2.878 9884.922 79 2.979 12862.45 80 3.08 16274.93 81 3.181 20156.15 82 3.282 24541.02 83 3.383 29465.6 84 3.484 34967.05 85 3.585 41083.69 86 3.686 47854.93 87 3.787 55321.34 88 3.888 63524.5889 3.989 72507.48

90 4.09 82313.95 91 4.191 92989.07 92 4.292 104579 93 4.393 117131.1 94 4.494 130693.8 95 4.595 145316.6 96 4.696 161050.3 97 4.797 177946.6 98 4.898 196058.6 99 4.999 215440.2distribution of Z Normal distribution

1.48672E-06 0.320584728 2.45094E-06 0.480968059 3.9995E-06 0.712368881 6.46025E-06 1.041434637 1.03291E-05 1.502490538 1.63472E-05 2.138691351 2.56093E-05 3.002845525 3.97118E-05 4.157611109 6.09553E-05 5.674674747 9.26132E-05 7.632445337 0.000139285 10.11174029 0.000207351 13.18893815 0.000305546 16.92614391 0.000445674 21.35809064 0.000643469 26.4758084 0.000919619 32.20754271 0.001300942 38.3979977 0.001821704 44.78768437 0.002525035 50.99491248 0.003464389 56.50367799 0.004704958 60.66123887 0.006324913 62.68938565 0.008416337 61.71313917 0.011085658 56.80970624 0.014453386 47.07890339 0.018652949 31.73391561 0.023828414 10.20830309 0.030130931 -17.72815034 0.03771375 -51.85545301 0.046725789 -91.4047783 0.057303789 -134.990404 0.069563239 -180.5875352 0.083588399 -225.5693993 0.099421884 -266.8128203 0.117054396 -300.8751322 0.136415346 -324.2372193 0.157365126 -333.5984632 0.179689839 -326.2005536 0.203099244 -300.1497382 0.227228529 -254.7024581 

# 

#  

2 4 2 

## 336 6 24 z 

## z z   

    

     

#   2 1 3 

### ( ) 1 3 ( )

6 24 z G 

### f z z z z             

## ( ) z 0.251644341 -190.4784967 0.275855243 -109.5694599 0.299326439 -15.51867645 0.321498297 86.83909555 0.341807853 191.6746972 0.359712192 292.5787914 0.37471238 383.0587505 0.386376486 457.0718861 0.394360216 509.5419173 0.398423793 536.8049788 0.398443914 536.9374517 0.394419966 509.9298876 0.386474058 457.6878688 0.374844865 383.8597112 0.359875719 293.5099852 0.34199778 192.6744437 0.32170943 87.84398392 0.299553264 -14.56868599 0.276092166 -108.7265328 0.25188591 -189.783421 0.227469632 -254.1824092 0.203335281 -299.8174229 0.17991684 -326.0546888 0.157579839 -333.6254108 0.136615273 -324.4134915 0.117237788 -301.1704294 0.099587708 -267.1932449 0.083736272 -226.0005213 0.069693339 -181.0370431 0.05741676 -135.4301614 0.046822635 -91.81218949 0.037795734 -52.21413206 0.030199481 -18.02795645 0.023885038 9.971748348 0.018699163 31.56008415 0.014490659 46.96343636 0.011115369 56.74556679 0.008439746 61.6917223 0.006343145 62.70149656 0.004718997 60.69788418 0.003475077 56.55664587 0.002533081 51.05714379 0.001827693 44.85345646 0.001305351 38.4629548 0.000922829 32.26861206 0.00064578 26.53104138 0.00044732 21.4064641 0.000306705 16.96735009 0.000208159 13.22318460.000139842 10.13957335 9.29928E-05 7.654604526 6.12113E-05 5.691980245 3.98826E-05 4.17088268 2.5722E-05 3.012849008 1.64209E-05 2.146107647 1.03767E-05 1.507901753 6.49066E-06 1.045322392 4.01874E-06 0.715120523 2.46298E-06 0.482887348 1.49417E-06 0.321904473Absolute values

0.320584728 0.016189529 0.480968059 0.048577774 0.712368881 0.071949257 1.041434637 0.105184898 1.502490538 0.151751544 2.138691351 0.216007826 3.002845525 0.303287398 4.157611109 0.419918722 5.674674747 0.573142149 7.632445337 0.770876979 10.11174029 1.021285769 13.18893815 1.332082753 16.92614391 1.709540535 21.35809064 2.157167155 26.4758084 2.674056648 32.20754271 3.252961814 38.3979977 3.878197768 44.78768437 4.523556122 50.99491248 5.150486161 56.50367799 5.706871477 60.66123887 6.126785126 62.68938565 6.331627951 61.71313917 6.233027056 56.80970624 5.737780331 47.07890339 4.754969242 31.73391561 3.205125477 10.20830309 1.031038612 17.72815034 1.790543185 51.85545301 5.237400755 91.4047783 9.231882608 134.990404 13.6340308 180.5875352 18.23934105 225.5693993 22.78250933 266.8128203 26.94809485 300.8751322 30.38838836 324.2372193 32.74795915 333.5984632 33.69344479 326.2005536 32.94625591 300.1497382 30.31512356 254.7024581 25.72494826 

## 

#  

2 4 2 

### 336 ( ) 1 3 ( )

6 24 z 

### z z f z z z z    

       1 

## ( ) 2 () () 2 M i

i 

## xf a f x f b

 

##  

      

# 190.4784967 19.23832816 109.5694599 11.06651545 15.51867645 1.567386321 86.83909555 8.77074865 191.6746972 19.35914441 292.5787914 29.55045794 383.0587505 38.6889338 457.0718861 46.16426049 509.5419173 51.46373365 536.8049788 54.21730286 536.9374517 54.23068262 509.9298876 51.50291865 457.6878688 46.22647475 383.8597112 38.76983083 293.5099852 29.6445085 192.6744437 19.46011881 87.84398392 8.872242376 14.56868599 1.471437285 108.7265328 10.98137981 189.783421 19.16812552 254.1824092 25.67242333 299.8174229 30.28155971 326.0546888 32.93152357 333.6254108 33.69616649 324.4134915 32.76576264 301.1704294 30.41821337 267.1932449 26.98651774 226.0005213 22.82605265 181.0370431 18.28474136 135.4301614 13.6784463 91.81218949 9.273031139 52.21413206 5.273627338 18.02795645 1.820823602 9.971748348 1.007146583 31.56008415 3.187568499 46.96343636 4.743307072 56.74556679 5.731302246 61.6917223 6.230863952 62.70149656 6.332851153 60.69788418 6.130486302 56.55664587 5.712221233 51.05714379 5.156771522 44.85345646 4.530199103 38.4629548 3.884758435 32.26861206 3.259129818 26.53104138 2.67963518 21.4064641 2.162052874 16.96735009 1.713702359 13.2231846 1.33554164510.13957335 1.024096908 7.654604526 0.773115057 5.691980245 0.574890005 4.17088268 0.421259151 3.012849008 0.30429775 2.146107647 0.216756872 1.507901753 0.152298077 1.045322392 0.105577562 0.715120523 0.072227173 0.482887348 0.048771622 0.321904473 0.016256176Cumulative normalization
0.016189529 1.28182E-05 0.064767303 5.128E-05 0.13671656 0.000108246 0.241901458 0.000191527 0.393653002 0.000311677 0.609660829 0.000482703 0.912948227 0.000722833 1.332866949 0.001055307 1.906009098 0.001509096 2.676886077 0.002119443 3.698171846 0.002928053 5.0302546 0.003982739 6.739795135 0.00533628 8.896962289 0.007044232 11.57101894 0.009161435 14.82398075 0.01173699 18.70217852 0.01480758 23.22573464 0.018389137 28.3762208 0.02246707 34.08309228 0.026985525 40.2098774 0.03183645 46.54150536 0.03684956 52.77453241 0.041784602 58.51231274 0.04632753 63.26728198 0.05009231 66.47240746 0.05262999 67.50344607 0.053446322 69.29398926 0.054863997 74.53139001 0.059010745 83.76327262 0.066320152 97.39730342 0.077114991 115.6366445 0.091556116 138.4191538 0.109594326 165.3672487 0.130930667 195.755637 0.154990884 228.5035962 0.180919308 262.1970409 0.207596327 295.1432969 0.233681754 325.4584204 0.257683963 351.1833687 0.27805187 

## ( ) 2 () () f a f x f b        370.4216968 0.293283949 381.4882123 0.30204594 383.0555986 0.303286929 391.8263473 0.31023123 411.1854917 0.325558967 440.7359496 0.348955747 479.4248834 0.379587979 525.5891439 0.416138853 577.0528776 0.456885621 631.2701804 0.499812547 685.500863 0.542750066 737.0037817 0.583527859 783.2302564 0.620127992 822.0000873 0.650824274 851.6445958 0.67429552 871.1047146 0.689703204 879.976957 0.696727863 881.4483942 0.697892884 892.429774 0.706587468 911.5978996 0.721763964 937.2703229 0.742090283 967.5518826 0.766065918 1000.483406 0.792139681 1034.179573 0.818818854 1066.945335 0.844761374 1097.363549 0.868845206 1124.350066 0.890211969 1147.176119 0.908284654 1165.46086 0.922761725 1179.139307 0.93359173 1188.412338 0.940933717 1193.685965 0.945109148 1195.506789 0.946550797 1196.513935 0.947348213 1199.701504 0.949871992 1204.444811 0.953627539 1210.176113 0.958165337 1216.406977 0.963098667 1222.739828 0.968112745 1228.870315 0.972966601 1234.582536 0.977489291 1239.739307 0.981572201 1244.269506 0.985159018 1248.154265 0.988234802 1251.413395 0.990815241 1254.09303 0.99293686 1256.255083 0.994648681 1257.968785 0.996005517 1259.304327 0.9970629411260.328424 0.997873777 1261.101539 0.998485896 1261.676429 0.998941069 1262.097688 0.999274604 1262.401986 0.999515534 1262.618743 0.999687153 1262.771041 0.999807736 1262.876618 0.999891327 1262.948845 0.999948514 1262.997617 0.999987129 1263.013873 1 63776.35047Raw Data rank by lb/ton use lb/hr Runs 568 1 1.63E-04 2 1.78E-04 3 1.85E-04 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 Testing Normality Raw Data Sample Size 3.00 Kurtosis #DIV/0! SE Kurtosis #DIV/0! Result Kurtosis #DIV/0! Skewness -1.00 SE Skewness 1.22 Result Raw Data Result Skewness normal #DIV/0!
ni = number test runs = 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Total Number of sources = 1 n =Total # test runs = 3

2
This step is for QC only: means 1.75E-04 #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

1.75E-04 Mean =
Pooled Variance = 1.25E-10

m= number future runs = 3 Term1 6.67E-01
Term2 8.31E-11

Squared root Term2 9.12E-06 NOTE: the pvalue for the t-statistic is calculated as: 2*alpha, where 1-alpha is desired confidence, so if 99% confidence is desired then alpha=0.01 and 2*alpha=2*(0.01) t-statistic = quantile t-distribution with df degrees of freedom at .99 confidence level = 6.964556734 99 UPL POOLED VARIANCE = 2.39E-04 Use for large sample sets and small sample sets that are Normal.

 2 2 1 11n i i s x X n   

 1 1 1 i

n N ij i j X x n   

1 1 n m 

21 1 sn m        1 1 N i i df n

         

df t 2 1 1

s n m        1 1 i n i i

k k i x x n  1

N i in

  N

2

, 1 1 dfp X t s n m    

   

0900006481934216 Hg-Large-lbhr 12/1/2014Raw Data rank by lb/ton use lb/ton Runs 531 1 1.84E-04 2 1.69E-04 3 1.76E-04 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 Testing Normality Raw Data Logtransformed Data Sample Size 3.00 3.00 Kurtosis #DIV/0! #DIV/0! SE Kurtosis #DIV/0! #DIV/0! Average of Raw Data 1.76E-04 Result Kurtosis #DIV/0! #DIV/0! Skewness 0.09 0.03 SE Skewness 1.22 1.22 Result Raw Data Result Log Data Result Skewness normal normal #DIV/0! #DIV/0! 1. Take LN( ) of Raw Data Runs 531 1 -8.60E+00 2 -8.69E+00 3 -8.64E+00 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 3

3. Calculate -8.64E+00

0.001773919 4. Calculate

5. m= number future runs = 3 6. Calculate Use for small sample sets that are Lognormal.

2. Calculate sample size, n=

1 ˆ n i

i x n     2 2 1 ˆ 1

n i i x x n   


 

### 

222 ˆˆˆ 4 3 2 233 1 3 1 e e e          

0900006481934216 Hg-Small-lbton 12/1/20143.028455212
9.45713E-06

320231.8612
7. Calculate 0.07302593

9. substitue the values of in cell D7 and F7 respectively

3.484z value from "z-stat"

a- Calculate 0.000176319
b- Calculate 3.10885E-08

c- Calculate 0.001775493

d-calculate 0.000592093

e- Calculate 1.82E-05

f- Calculate UPL 2.0E-04 11. In column B, go down the the row number from Step 10. above, and copy the z value. 12. calculate the UPL using the formula a- Numerator first term
b- denominator first term

8. Go to tab distribution for 99 percentile "z-stat"

10. In column AE indentify the value that is the smallest value that is larger than 0.99, note what row number this is. 222 ˆˆˆ 4 3 2 233 e e e   

###   2 2 ˆ 1 m e



2z    2 2 ˆ ˆ 1 1 2
z e e

m      

2z

 1z 

2 ˆ 2 2 2 2 2 4
ˆ ˆ ˆ ˆ ˆ ˆ 2 2 2 .99 ˆ ˆ

( 1) 2( 1) z UPL e me e m e

m n n                    

  

2 ˆ 2 ˆ e  
2 ˆ ˆ 2 e  

2 ˆ ( 1) e



2 4 ˆ ˆ 2( 1) n n       

  

2 2 2 2 4
ˆ ˆ ˆ ˆ ˆ 2 2 2 ˆ ˆ ( 1) 2( 1) me e me n n             

 

     2 2 2 ˆ 233 1 3 1 1 z e e e m

m e           



0900006481934216 Hg-Small-lbton 12/1/2014∆z z

0.101 320231.9 0.073026

-5=f(a) 6377891 1 -4.899 5804260 2 -4.798 5268199 3 -4.697 4768108 4 -4.596 4302421 5 -4.495 3869604 6 -4.394 3468158 7 -4.293 3096616 8 -4.192 2753546 9 -4.091 2437548 10 -3.99 2147255 11 -3.889 1881334 12 -3.788 1638485 13 -3.687 1417443 14 -3.586 1216973 15 -3.485 1035876 16 -3.384 872986 17 -3.283 727169.3 18 -3.182 597326.2 19 -3.081 482390.2 20 -2.98 381328 21 -2.879 293140 22 -2.778 216859.4 23 -2.677 151553.1 24 -2.576 96321.29 25 -2.475 50297.27 26 -2.374 12647.86 27 -2.273 -17426.9 28 -2.172 -40693.5 29 -2.071 -57885.4 30 -1.97 -69702.4 31 -1.869 -76811.1 32 -1.768 -79845 33 -1.667 -79403.9 34 -1.566 -76054.5 35 -1.465 -70330.2 36 -1.364 -62730.9 37 -1.263 -53723.4 38 -1.162 -43741 39 -1.061 -33183.7 

#   1 3 

## 1 3 6 24 z 

## z z      

     2z  1z



## 40 -0.96 -22418.4

41 -0.859 -11778.3 42 -0.758 -1563.57 43 -0.657 7959.066 44 -0.556 16556.17 45 -0.455 24027.64 46 -0.354 30206.71 47 -0.253 34959.9 48 -0.152 38187.09 49 -0.051 39821.47 50 0.05 39829.55 51 0.151 38211.15 52 0.252 34999.45 53 0.353 30260.93 54 0.454 24095.38 55 0.555 16635.93 56 0.656 8049.039 57 0.757 -1465.52 58 0.858 -11674.7 59 0.959 -22311.9 60 1.06 -33077.6 61 1.161 -43638.7 62 1.262 -53628.7 63 1.363 -62647.9 64 1.464 -70263.3 65 1.565 -76008.6 66 1.666 -79384.1 67 1.767 -79856.7 68 1.868 -76860.1 69 1.969 -69794.7 70 2.07 -58027.5 71 2.171 -40892.3 72 2.272 -17689.3 73 2.373 12314.41 74 2.474 49884.99 75 2.575 95822.1 76 2.676 150958.6 77 2.777 216160.8 78 2.878 292328.3 79 2.979 380393.8 80 3.08 481323.7 81 3.181 596117.4 82 3.282 725807.8 83 3.383 871461.1 84 3.484 1034177 85 3.585 1215088 86 3.686 1415360 87 3.787 1636193 88 3.888 187882089 3.989 2144506

90 4.09 2434551 91 4.191 2750289 92 4.292 3093084 93 4.393 3464337 94 4.494 3865480 95 4.595 4297979 96 4.696 4763334 97 4.797 5263077 98 4.898 5798774 99 4.999 6372025distribution of Z Normal distribution

1.48672E-06 9.482135231 2.45094E-06 14.22589281 3.9995E-06 21.07017796 6.46025E-06 30.80315723 1.03291E-05 44.44008603 1.63472E-05 63.25737464 2.56093E-05 88.8169625 3.97118E-05 122.9721103 6.09553E-05 167.8431131 9.26132E-05 225.7490785 0.000139285 299.0803286 0.000207351 390.0958602 0.000305546 500.6324415 0.000445674 631.7171745 0.000643469 783.0844331 0.000919619 952.6114436 0.001300942 1135.704304 0.001821704 1324.687107 0.002525035 1508.269244 0.003464389 1671.187086 0.004704958 1794.132213 0.006324913 1854.084705 0.008416337 1825.161903 0.011085658 1680.066356 0.014453386 1392.168772 0.018652949 938.1924676 0.023828414 301.3784363 0.030130931 -525.0880094 0.03771375 -1534.705674 0.046725789 -2704.740072 0.057303789 -3994.210046 0.069563239 -5343.231447 0.083588399 -6674.1139 0.099421884 -7894.483006 0.117054396 -8902.512616 0.136415346 -9594.112976 0.157365126 -9871.654295 0.179689839 -9653.544101 0.203099244 -8883.758627 0.227228529 -7540.291364 

# 

#  

2 4 2 

## 336 6 24 z 

## z z   

    

     

#   2 1 3 

### ( ) 1 3 ( )

6 24 z G 

### f z z z z             

## ( ) z 0.251644341 -5641.457864 0.275855243 -3249.104849 0.299326439 -468.0164235 0.321498297 2558.8262 0.341807853 5659.029172 0.359712192 8643.036797 0.37471238 11318.82738 0.386376486 13507.68414 0.394360216 15059.47049 0.398423793 15865.82151 0.398443914 15869.84028 0.394419966 15071.24222 0.386474058 13526.38094 0.374844865 11343.15265 0.359875719 8671.340423 0.34199778 5689.45084 0.32170943 2589.451717 0.299553264 -439.0018051 0.276092166 -3223.280932 0.25188591 -5620.06389 0.227469632 -7524.158318 0.203335281 -8873.282863 0.17991684 -9648.701009 0.157579839 -9872.046658 0.136615273 -9599.045599 0.117237788 -8911.083046 0.099587708 -7905.677738 0.083736272 -6686.900326 0.069693339 -5356.637602 0.05741676 -4007.386645 0.046822635 -2717.001999 0.037795734 -1545.55264 0.030199481 -534.2062408 0.023885038 294.1300802 0.018699163 932.807623 0.014490659 1388.525396 0.011115369 1677.960975 0.008439746 1824.342699 0.006343145 1854.280771 0.004718997 1795.0773 0.003475077 1672.637104 0.002533081 1510.013548 0.001827693 1326.554202 0.001305351 1137.562967 0.000922829 954.3684071 0.00064578 784.6798087 0.00044732 633.1186513 0.000306705 501.829117 0.000208159 391.09233490.000139842 299.8914797 9.29928E-05 226.395735 6.12113E-05 168.3487008 3.98826E-05 123.3602238 2.5722E-05 89.10975263 1.64209E-05 63.47460261 1.03767E-05 44.59868829 6.49066E-06 30.91717381 4.01874E-06 21.15091801 2.46298E-06 14.28223616 1.49417E-06 9.520894664Absolute values

9.482135231 0.478847829 14.22589281 1.436815174 21.07017796 2.128087974 30.80315723 3.11111888 44.44008603 4.488448689 63.25737464 6.388994839 88.8169625 8.970513213 122.9721103 12.42018314 167.8431131 16.95215442 225.7490785 22.80065693 299.0803286 30.20711319 390.0958602 39.39968188 500.6324415 50.56387659 631.7171745 63.80343462 783.0844331 79.09152774 952.6114436 96.21375581 1135.704304 114.7061347 1324.687107 133.7933978 1508.269244 152.3351937 1671.187086 168.7898957 1794.132213 181.2073535 1854.084705 187.2625552 1825.161903 184.3413522 1680.066356 169.6867019 1392.168772 140.609046 938.1924676 94.75743923 301.3784363 30.43922207 525.0880094 53.03388895 1534.705674 155.005273 2704.740072 273.1787473 3994.210046 403.4152146 5343.231447 539.6663761 6674.1139 674.0855039 7894.483006 797.3427836 8902.512616 899.1537742 9594.112976 969.0054106 9871.654295 997.0370838 9653.544101 975.0079542 8883.758627 897.2596213 7540.291364 761.5694277 

## 

#  

2 4 2 

### 336 ( ) 1 3 ( )

6 24 z 

### z z f z z z z    

       1 

## ( ) 2 () () 2 M i

i 

## xf a f x f b

 

##  

      

# 5641.457864 569.7872442 3249.104849 328.1595898 468.0164235 47.26965877 2558.8262 258.4414462 5659.029172 571.5619464 8643.036797 872.9467164 11318.82738 1143.201565 13507.68414 1364.276098 15059.47049 1521.006519 15865.82151 1602.447973 15869.84028 1602.853868 15071.24222 1522.195464 13526.38094 1366.164475 11343.15265 1145.658418 8671.340423 875.8053828 5689.45084 574.6345348 2589.451717 261.5346234 439.0018051 44.33918231 3223.280932 325.5513742 5620.06389 567.6264529 7524.158318 759.9399901 8873.282863 896.2015692 9648.701009 974.5188019 9872.046658 997.0767124 9599.045599 969.5036055 8911.083046 900.0193876 7905.677738 798.4734515 6686.900326 675.3769329 5356.637602 541.0203978 4007.386645 404.7460512 2717.001999 274.4172019 1545.55264 156.1008167 534.2062408 53.95483033 294.1300802 29.7071381 932.807623 94.21356992 1388.525396 140.241065 1677.960975 169.4740585 1824.342699 184.2586126 1854.280771 187.2823578 1795.0773 181.3028073 1672.637104 168.9363475 1510.013548 152.5113683 1326.554202 133.9819744 1137.562967 114.8938597 954.3684071 96.39120912 784.6798087 79.25266068 633.1186513 63.94498378 501.829117 50.68474082 391.0923349 39.50032582299.8914797 30.28903945 226.395735 22.86596924 168.3487008 17.00321878 123.3602238 12.4593826 89.10975263 9.000085016 63.47460261 6.410934864 44.59868829 4.504467517 30.91717381 3.122634555 21.15091801 2.136242719 14.28223616 1.442505852 9.520894664 0.480805181Cumulative normalization
0.478847829 1.28191E-05 1.915663003 5.12837E-05 4.043750977 0.000108254 7.154869857 0.000191541 11.64331855 0.0003117 18.03231338 0.000482738 27.0028266 0.000722885 39.42300974 0.001055382 56.37516416 0.001509204 79.17582108 0.002119593 109.3829343 0.002928259 148.7826162 0.003983017 199.3464927 0.005336648 263.1499274 0.007044712 342.2414551 0.009162049 438.4552109 0.01173776 553.1613456 0.014808525 686.9547434 0.01839027 839.2899371 0.022468393 1008.079833 0.026987019 1189.287186 0.03183807 1376.549741 0.036851222 1560.891094 0.041786172 1730.577796 0.046328807 1871.186842 0.050093012 1965.944281 0.052629736 1996.383503 0.053444616 2049.417392 0.054864371 2204.422665 0.059013973 2477.601412 0.066327164 2881.016627 0.077126878 3420.683003 0.091574133 4094.768507 0.109619885 4892.11129 0.130965322 5791.265065 0.155036313 6760.270475 0.180977282 7757.307559 0.207668678 8732.315513 0.233770339 9629.575134 0.257790622 10391.14456 0.278178381 

## ( ) 2 () () f a f x f b        10960.93181 0.293431993 11289.0914 0.302217061 11336.36105 0.303482503 11594.8025 0.310401166 12166.36445 0.325702288 13039.31116 0.349071697 14182.51273 0.379676022 15546.78883 0.41619867 17067.79535 0.456917103 18670.24332 0.499815783 20273.09719 0.542725329 21795.29265 0.583475591 23161.45713 0.620048793 24307.11554 0.650718889 25182.92093 0.674164826 25757.55546 0.689548204 26019.09009 0.696549673 26063.42927 0.697736665 26388.98064 0.706451908 26956.6071 0.721647675 27716.54709 0.741991813 28612.74865 0.765983771 29587.26746 0.792072337 30584.34417 0.818764794 31553.84777 0.8447191 32453.86716 0.868813264 33252.34061 0.89018897 33927.71755 0.908269294 34468.73794 0.922752797 34873.484 0.933588139 35147.9012 0.940934484 35304.00201 0.945113414 35357.95684 0.946557824 35387.66398 0.947353105 35481.87755 0.94987527 35622.11862 0.953629624 35791.59268 0.958166566 35975.85129 0.963099301 36163.13365 0.968112983 36344.43645 0.972966589 36513.3728 0.977489136 36665.88417 0.981571975 36799.86614 0.985158768 36914.76 0.988234559 37011.15121 0.990815021 37090.40387 0.992936671 37154.34886 0.994648524 37205.0336 0.996005391 37244.53392 0.99706284337274.82296 0.997873703 37297.68893 0.998485841 37314.69215 0.998941029 37327.15153 0.999274576 37336.15162 0.999515515 37342.56255 0.99968714 37347.06702 0.999807728 37350.18966 0.999891323 37352.3259 0.999948512 37353.7684 0.999987129 37354.24921 1 1886224.019Raw Data rank by lb/ton using concentration data Runs 531 1 6.03E+01 2 6.21E+01 3 6.32E+01 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 Testing Normality Raw Data Sample Size 3.00 Kurtosis #DIV/0! SE Kurtosis #DIV/0! Result Kurtosis #DIV/0! Skewness -0.60 SE Skewness 1.22 Result Raw Data Result Skewness normal #DIV/0!
ni = number test runs = 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Total Number of sources = 1 n =Total # test runs = 3

2
This step is for QC only: means 6.19E+01 #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

6.19E+01 Mean =
Pooled Variance = 2.18E+00

m= number future runs = 3 Term1 6.67E-01
Term2 1.45E+00

Squared root Term2 1.21E+00 NOTE: the pvalue for the t-statistic is calculated as: 2*alpha, where 1-alpha is desired confidence, so if 99% confidence is desired then alpha=0.01 and 2*alpha=2*(0.01) t-statistic = quantile t-distribution with df degrees of freedom at .99 confidence level = 6.964556734 99 UPL POOLED VARIANCE = 70 Use for large sample sets and small sample sets that are Normal.

 2 2 1 11n i i s x X n   

 1 1 1 i

n N ij i j X x n   

1 1 n m 

21 1 sn m        1 1 N i i df n

         

df t 2 1 1

s n m        1 1 i n i i

k k i x x n  1

N i in

  N

2

, 1 1 dfp X t s n m    

   

0900006481934216 Hg-Small-conc 12/1/2014Raw Data rank by lb/ton using lb/hr data Runs 531 1 1.04E-03 2 9.55E-04 3 9.96E-04 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 Testing Normality Raw Data Logtransformed Data Sample Size 3.00 3.00 Kurtosis #DIV/0! #DIV/0! SE Kurtosis #DIV/0! #DIV/0! Average of Raw Data 9.97E-04 Result Kurtosis #DIV/0! #DIV/0! Skewness 0.09 0.03 SE Skewness 1.22 1.22 Result Raw Data Result Log Data Result Skewness normal normal #DIV/0! #DIV/0! 1. Take LN( ) of Raw Data Runs 531 1 -6.87E+00 2 -6.95E+00 3 -6.91E+00 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 3

3. Calculate -6.91E+00

0.001773919 4. Calculate

5. m= number future runs = 3 6. Calculate Use for small sample sets that are Lognormal.

2. Calculate sample size, n=

1 ˆ n i

i x n     2 2 1 ˆ 1

n i i x x n   


 

### 

222 ˆˆˆ 4 3 2 233 1 3 1 e e e          

0900006481934216 Hg-Small-lbhr 12/1/20143.028455212
9.45713E-06

320231.8612
7. Calculate 0.07302593

9. substitue the values of in cell D7 and F7 respectively

3.484z value from "z-stat"

a- Calculate 0.000997086
b- Calculate 9.9418E-07

c- Calculate 0.001775493

d-calculate 0.000592093

e- Calculate 0.000103

f- Calculate UPL 0.0011 11. In column B, go down the the row number from Step 10. above, and copy the z value. 12. calculate the UPL using the formula a- Numerator first term
b- denominator first term

8. Go to tab distribution for 99 percentile "z-stat"

10. In column AE indentify the value that is the smallest value that is larger than 0.99, note what row number this is. 222 ˆˆˆ 4 3 2 233 e e e   

###   2 2 ˆ 1 m e



2z    2 2 ˆ ˆ 1 1 2
z e e

m      

2z

 1z 

2 ˆ 2 2 2 2 2 4
ˆ ˆ ˆ ˆ ˆ ˆ 2 2 2 .99 ˆ ˆ

( 1) 2( 1) z UPL e me e m e

m n n                    

  

2 ˆ 2 ˆ e  
2 ˆ ˆ 2 e  

2 ˆ ( 1) e



2 4 ˆ ˆ 2( 1) n n       

  

2 2 2 2 4
ˆ ˆ ˆ ˆ ˆ 2 2 2 ˆ ˆ ( 1) 2( 1) me e me n n             

 

     2 2 2 ˆ 233 1 3 1 1 z e e e m

m e           



0900006481934216 Hg-Small-lbhr 12/1/2014∆z z

0.101 320231.9 0.073026

-5=f(a) 6377891 1 -4.899 5804260 2 -4.798 5268199 3 -4.697 4768108 4 -4.596 4302421 5 -4.495 3869604 6 -4.394 3468158 7 -4.293 3096616 8 -4.192 2753546 9 -4.091 2437548 10 -3.99 2147255 11 -3.889 1881334 12 -3.788 1638485 13 -3.687 1417443 14 -3.586 1216973 15 -3.485 1035876 16 -3.384 872986 17 -3.283 727169.3 18 -3.182 597326.2 19 -3.081 482390.2 20 -2.98 381328 21 -2.879 293140 22 -2.778 216859.4 23 -2.677 151553.1 24 -2.576 96321.29 25 -2.475 50297.27 26 -2.374 12647.86 27 -2.273 -17426.9 28 -2.172 -40693.5 29 -2.071 -57885.4 30 -1.97 -69702.4 31 -1.869 -76811.1 32 -1.768 -79845 33 -1.667 -79403.9 34 -1.566 -76054.5 35 -1.465 -70330.2 36 -1.364 -62730.9 37 -1.263 -53723.4 38 -1.162 -43741 39 -1.061 -33183.7 

#   1 3 

## 1 3 6 24 z 

## z z      

     2z  1z



## 40 -0.96 -22418.4

41 -0.859 -11778.3 42 -0.758 -1563.57 43 -0.657 7959.066 44 -0.556 16556.17 45 -0.455 24027.64 46 -0.354 30206.71 47 -0.253 34959.9 48 -0.152 38187.09 49 -0.051 39821.47 50 0.05 39829.55 51 0.151 38211.15 52 0.252 34999.45 53 0.353 30260.93 54 0.454 24095.38 55 0.555 16635.93 56 0.656 8049.039 57 0.757 -1465.52 58 0.858 -11674.7 59 0.959 -22311.9 60 1.06 -33077.6 61 1.161 -43638.7 62 1.262 -53628.7 63 1.363 -62647.9 64 1.464 -70263.3 65 1.565 -76008.6 66 1.666 -79384.1 67 1.767 -79856.7 68 1.868 -76860.1 69 1.969 -69794.7 70 2.07 -58027.5 71 2.171 -40892.3 72 2.272 -17689.3 73 2.373 12314.41 74 2.474 49884.99 75 2.575 95822.1 76 2.676 150958.6 77 2.777 216160.8 78 2.878 292328.3 79 2.979 380393.8 80 3.08 481323.7 81 3.181 596117.4 82 3.282 725807.8 83 3.383 871461.1 84 3.484 1034177 85 3.585 1215088 86 3.686 1415360 87 3.787 1636193 88 3.888 187882089 3.989 2144506

90 4.09 2434551 91 4.191 2750289 92 4.292 3093084 93 4.393 3464337 94 4.494 3865480 95 4.595 4297979 96 4.696 4763334 97 4.797 5263077 98 4.898 5798774 99 4.999 6372025distribution of Z Normal distribution

1.48672E-06 9.482135231 2.45094E-06 14.22589281 3.9995E-06 21.07017796 6.46025E-06 30.80315723 1.03291E-05 44.44008603 1.63472E-05 63.25737464 2.56093E-05 88.8169625 3.97118E-05 122.9721103 6.09553E-05 167.8431131 9.26132E-05 225.7490785 0.000139285 299.0803286 0.000207351 390.0958602 0.000305546 500.6324415 0.000445674 631.7171745 0.000643469 783.0844331 0.000919619 952.6114436 0.001300942 1135.704304 0.001821704 1324.687107 0.002525035 1508.269244 0.003464389 1671.187086 0.004704958 1794.132213 0.006324913 1854.084705 0.008416337 1825.161903 0.011085658 1680.066356 0.014453386 1392.168772 0.018652949 938.1924676 0.023828414 301.3784363 0.030130931 -525.0880094 0.03771375 -1534.705674 0.046725789 -2704.740072 0.057303789 -3994.210046 0.069563239 -5343.231447 0.083588399 -6674.1139 0.099421884 -7894.483006 0.117054396 -8902.512616 0.136415346 -9594.112976 0.157365126 -9871.654295 0.179689839 -9653.544101 0.203099244 -8883.758627 0.227228529 -7540.291364 

# 

#  

2 4 2 

## 336 6 24 z 

## z z   

    

     

#   2 1 3 

### ( ) 1 3 ( )

6 24 z G 

### f z z z z             

## ( ) z 0.251644341 -5641.457864 0.275855243 -3249.104849 0.299326439 -468.0164235 0.321498297 2558.8262 0.341807853 5659.029172 0.359712192 8643.036797 0.37471238 11318.82738 0.386376486 13507.68414 0.394360216 15059.47049 0.398423793 15865.82151 0.398443914 15869.84028 0.394419966 15071.24222 0.386474058 13526.38094 0.374844865 11343.15265 0.359875719 8671.340423 0.34199778 5689.45084 0.32170943 2589.451717 0.299553264 -439.0018051 0.276092166 -3223.280932 0.25188591 -5620.06389 0.227469632 -7524.158318 0.203335281 -8873.282863 0.17991684 -9648.701009 0.157579839 -9872.046658 0.136615273 -9599.045599 0.117237788 -8911.083046 0.099587708 -7905.677738 0.083736272 -6686.900326 0.069693339 -5356.637602 0.05741676 -4007.386645 0.046822635 -2717.001999 0.037795734 -1545.55264 0.030199481 -534.2062408 0.023885038 294.1300802 0.018699163 932.807623 0.014490659 1388.525396 0.011115369 1677.960975 0.008439746 1824.342699 0.006343145 1854.280771 0.004718997 1795.0773 0.003475077 1672.637104 0.002533081 1510.013548 0.001827693 1326.554202 0.001305351 1137.562967 0.000922829 954.3684071 0.00064578 784.6798087 0.00044732 633.1186513 0.000306705 501.829117 0.000208159 391.09233490.000139842 299.8914797 9.29928E-05 226.395735 6.12113E-05 168.3487008 3.98826E-05 123.3602238 2.5722E-05 89.10975263 1.64209E-05 63.47460261 1.03767E-05 44.59868829 6.49066E-06 30.91717381 4.01874E-06 21.15091801 2.46298E-06 14.28223616 1.49417E-06 9.520894664Absolute values

9.482135231 0.478847829 14.22589281 1.436815174 21.07017796 2.128087974 30.80315723 3.11111888 44.44008603 4.488448689 63.25737464 6.388994839 88.8169625 8.970513213 122.9721103 12.42018314 167.8431131 16.95215442 225.7490785 22.80065693 299.0803286 30.20711319 390.0958602 39.39968188 500.6324415 50.56387659 631.7171745 63.80343462 783.0844331 79.09152774 952.6114436 96.21375581 1135.704304 114.7061347 1324.687107 133.7933978 1508.269244 152.3351937 1671.187086 168.7898957 1794.132213 181.2073535 1854.084705 187.2625552 1825.161903 184.3413522 1680.066356 169.6867019 1392.168772 140.609046 938.1924676 94.75743923 301.3784363 30.43922207 525.0880094 53.03388895 1534.705674 155.005273 2704.740072 273.1787473 3994.210046 403.4152146 5343.231447 539.6663761 6674.1139 674.0855039 7894.483006 797.3427836 8902.512616 899.1537742 9594.112976 969.0054106 9871.654295 997.0370838 9653.544101 975.0079542 8883.758627 897.2596213 7540.291364 761.5694277 

## 

#  

2 4 2 

### 336 ( ) 1 3 ( )

6 24 z 

### z z f z z z z    

       1 

## ( ) 2 () () 2 M i

i 

## xf a f x f b

 

##  

      

# 5641.457864 569.7872442 3249.104849 328.1595898 468.0164235 47.26965877 2558.8262 258.4414462 5659.029172 571.5619464 8643.036797 872.9467164 11318.82738 1143.201565 13507.68414 1364.276098 15059.47049 1521.006519 15865.82151 1602.447973 15869.84028 1602.853868 15071.24222 1522.195464 13526.38094 1366.164475 11343.15265 1145.658418 8671.340423 875.8053828 5689.45084 574.6345348 2589.451717 261.5346234 439.0018051 44.33918231 3223.280932 325.5513742 5620.06389 567.6264529 7524.158318 759.9399901 8873.282863 896.2015692 9648.701009 974.5188019 9872.046658 997.0767124 9599.045599 969.5036055 8911.083046 900.0193876 7905.677738 798.4734515 6686.900326 675.3769329 5356.637602 541.0203978 4007.386645 404.7460512 2717.001999 274.4172019 1545.55264 156.1008167 534.2062408 53.95483033 294.1300802 29.7071381 932.807623 94.21356992 1388.525396 140.241065 1677.960975 169.4740585 1824.342699 184.2586126 1854.280771 187.2823578 1795.0773 181.3028073 1672.637104 168.9363475 1510.013548 152.5113683 1326.554202 133.9819744 1137.562967 114.8938597 954.3684071 96.39120912 784.6798087 79.25266068 633.1186513 63.94498378 501.829117 50.68474082 391.0923349 39.50032582299.8914797 30.28903945 226.395735 22.86596924 168.3487008 17.00321878 123.3602238 12.4593826 89.10975263 9.000085016 63.47460261 6.410934864 44.59868829 4.504467517 30.91717381 3.122634555 21.15091801 2.136242719 14.28223616 1.442505852 9.520894664 0.480805181Cumulative normalization
0.478847829 1.28191E-05 1.915663003 5.12837E-05 4.043750977 0.000108254 7.154869857 0.000191541 11.64331855 0.0003117 18.03231338 0.000482738 27.0028266 0.000722885 39.42300974 0.001055382 56.37516416 0.001509204 79.17582108 0.002119593 109.3829343 0.002928259 148.7826162 0.003983017 199.3464927 0.005336648 263.1499274 0.007044712 342.2414551 0.009162049 438.4552109 0.01173776 553.1613456 0.014808525 686.9547434 0.01839027 839.2899371 0.022468393 1008.079833 0.026987019 1189.287186 0.03183807 1376.549741 0.036851222 1560.891094 0.041786172 1730.577796 0.046328807 1871.186842 0.050093012 1965.944281 0.052629736 1996.383503 0.053444616 2049.417392 0.054864371 2204.422665 0.059013973 2477.601412 0.066327164 2881.016627 0.077126878 3420.683003 0.091574133 4094.768507 0.109619885 4892.11129 0.130965322 5791.265065 0.155036313 6760.270475 0.180977282 7757.307559 0.207668678 8732.315513 0.233770339 9629.575134 0.257790622 10391.14456 0.278178381 

## ( ) 2 () () f a f x f b        10960.93181 0.293431993 11289.0914 0.302217061 11336.36105 0.303482503 11594.8025 0.310401166 12166.36445 0.325702288 13039.31116 0.349071697 14182.51273 0.379676022 15546.78883 0.41619867 17067.79535 0.456917103 18670.24332 0.499815783 20273.09719 0.542725329 21795.29265 0.583475591 23161.45713 0.620048793 24307.11554 0.650718889 25182.92093 0.674164826 25757.55546 0.689548204 26019.09009 0.696549673 26063.42927 0.697736665 26388.98064 0.706451908 26956.6071 0.721647675 27716.54709 0.741991813 28612.74865 0.765983771 29587.26746 0.792072337 30584.34417 0.818764794 31553.84777 0.8447191 32453.86716 0.868813264 33252.34061 0.89018897 33927.71755 0.908269294 34468.73794 0.922752797 34873.484 0.933588139 35147.9012 0.940934484 35304.00201 0.945113414 35357.95684 0.946557824 35387.66398 0.947353105 35481.87755 0.94987527 35622.11862 0.953629624 35791.59268 0.958166566 35975.85129 0.963099301 36163.13365 0.968112983 36344.43645 0.972966589 36513.3728 0.977489136 36665.88417 0.981571975 36799.86614 0.985158768 36914.76 0.988234559 37011.15121 0.990815021 37090.40387 0.992936671 37154.34886 0.994648524 37205.0336 0.996005391 37244.53392 0.99706284337274.82296 0.997873703 37297.68893 0.998485841 37314.69215 0.998941029 37327.15153 0.999274576 37336.15162 0.999515515 37342.56255 0.99968714 37347.06702 0.999807728 37350.18966 0.999891323 37352.3259 0.999948512 37353.7684 0.999987129 37354.24921 1 1886224.019Raw Data Runs 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 Testing Normality Raw Data Sample Size 0.00 Kurtosis #DIV/0! SE Kurtosis 0.00 Result Kurtosis #DIV/0! Skewness #DIV/0! SE Skewness 0.00 Result Raw Data Result Skewness #DIV/0! #DIV/0!
ni = number test runs = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Total Number of sources = 0 n =Total # test runs = 0

-1
This step is for QC only: means #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

#DIV/0! Mean =
Pooled Variance = #DIV/0!

m= number future runs = 3 Term1 #DIV/0!
Term2 #DIV/0!

Squared root Term2 #DIV/0! NOTE: the pvalue for the t-statistic is calculated as: 2*alpha, where 1-alpha is desired confidence, so if 99% confidence is desired then alpha=0.01 and 2*alpha=2*(0.01) t-statistic = quantile t-distribution with df degrees of freedom at .99 confidence level = #NUM! UPL POOLED VARIANCE = #DIV/0! Use for large sample sets and small sample sets that are Normal.

 2 2 1 11n i i s x X n   

 1 1 1 i

n N ij i j X x n   

1 1 n m 

21 1 sn m        1 1 N i i df n

         

df t 2 1 1

s n m        1 1 i n i i

k k i x x n  1

N i in

  N

2

, 1 1 dfp X t s n m    

   

0900006481934216 UPL Pooled Template 12/1/2014Raw Data Runs 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 Testing Normality Raw Data Logtransformed Data Sample Size 0.00 0.00 Kurtosis #DIV/0! #DIV/0! SE Kurtosis 0.00 0.00 Average of Raw Data #DIV/0! Result Kurtosis #DIV/0! #DIV/0! Skewness #DIV/0! #DIV/0! SE Skewness 0.00 0.00 Result Raw Data Result Log Data Result Skewness #DIV/0! #DIV/0! #DIV/0! #DIV/0! 1. Take LN( ) of Raw Data Runs 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 0

3. Calculate #DIV/0!

#DIV/0! 4. Calculate

5. m= number future runs = 3 6. Calculate Use for small sample sets that are Lognormal.

2. Calculate sample size, n=

1 ˆ n i

i x n     2 2 1 ˆ 1

n i i x x n   


 

### 

222 ˆˆˆ 4 3 2 233 1 3 1 e e e          

0900006481934216 Lognormal Template 12/1/2014#DIV/0! #DIV/0!
#DIV/0!
7. Calculate #DIV/0!

9. substitue the values of in cell D7 and F7 respectively

3.484z value from "z-stat"

a- Calculate #DIV/0!
b- Calculate #DIV/0!

c- Calculate #DIV/0!

d-calculate #DIV/0!

e- Calculate #DIV/0!

f- Calculate UPL #DIV/0! 11. In column B, go down the the row number from Step 10. above, and copy the z value. 12. calculate the UPL using the formula a- Numerator first term
b- denominator first term

8. Go to tab distribution for 99 percentile "z-stat"

10. In column AE indentify the value that is the smallest value that is larger than 0.99, note what row number this is. 222 ˆˆˆ 4 3 2 233 e e e   

###   2 2 ˆ 1 m e



2z    2 2 ˆ ˆ 1 1 2
z e e

m      

2z

 1z 

2 ˆ 2 2 2 2 2 4
ˆ ˆ ˆ ˆ ˆ ˆ 2 2 2 .99 ˆ ˆ

( 1) 2( 1) z UPL e me e m e

m n n                    

  

2 ˆ 2 ˆ e  
2 ˆ ˆ 2 e  

2 ˆ ( 1) e



2 4 ˆ ˆ 2( 1) n n       

  

2 2 2 2 4
ˆ ˆ ˆ ˆ ˆ 2 2 2 ˆ ˆ ( 1) 2( 1) me e me n n             

 

     2 2 2 ˆ 233 1 3 1 1 z e e e m

m e           



0900006481934216 Lognormal Template 12/1/2014∆z z

0.101 14.78551 3.387135

-5=f(a) 173.6307 1 -4.899 156.5385 2 -4.798 140.6598 3 -4.697 125.9391 4 -4.596 112.3223 5 -4.495 99.75638 6 -4.394 88.18975 7 -4.293 77.57188 8 -4.192 67.85352 9 -4.091 58.98665 10 -3.99 50.92447 11 -3.889 43.62141 12 -3.788 37.03312 13 -3.687 31.11647 14 -3.586 25.82959 15 -3.485 21.13179 16 -3.384 16.98365 17 -3.283 13.34694 18 -3.182 10.18468 19 -3.081 7.461103 20 -2.98 5.141681 21 -2.879 3.193102 22 -2.778 1.583284 23 -2.677 0.281372 24 -2.576 -0.74226 25 -2.475 -1.51602 26 -2.374 -2.06709 27 -2.273 -2.42141 28 -2.172 -2.6037 29 -2.071 -2.63747 30 -1.97 -2.54498 31 -1.869 -2.34728 32 -1.768 -2.06419 33 -1.667 -1.71429 34 -1.566 -1.31496 35 -1.465 -0.88233 36 -1.364 -0.43132 37 -1.263 0.024392 38 -1.162 0.472344 39 -1.061 0.901303 

#   1 3 

## 1 3 6 24 z 

## z z      

     2z  1z



## 40 -0.96 1.301261

41 -0.859 1.663439 42 -0.758 1.980282 43 -0.657 2.245462 44 -0.556 2.45388 45 -0.455 2.601659 46 -0.354 2.686152 47 -0.253 2.705937 48 -0.152 2.660818 49 -0.051 2.551826 50 0.05 2.381219 51 0.151 2.152479 52 0.252 1.870318 53 0.353 1.540671 54 0.454 1.170702 55 0.555 0.7688 56 0.656 0.344581 57 0.757 -0.09111 58 0.858 -0.52622 59 0.959 -0.94743 60 1.06 -1.34023 61 1.161 -1.68888 62 1.262 -1.97639 63 1.363 -2.18458 64 1.464 -2.294 65 1.565 -2.28401 66 1.666 -2.13273 67 1.767 -1.81705 68 1.868 -1.31264 69 1.969 -0.59394 70 2.07 0.365836 71 2.171 1.594699 72 2.272 3.121887 73 2.373 4.977865 74 2.474 7.194324 75 2.575 9.804181 76 2.676 12.84158 77 2.777 16.34189 78 2.878 20.34171 79 2.979 24.87887 80 3.08 29.9924 81 3.181 35.7226 82 3.282 42.11095 83 3.383 49.20019 84 3.484 57.03427 85 3.585 65.65838 86 3.686 75.1189387 3.787 85.46354

88 3.888 96.74107 89 3.989 109.0016 90 4.09 122.2965 91 4.191 136.6783 92 4.292 152.2006 93 4.393 168.9187 94 4.494 186.8885 95 4.595 206.1677 96 4.696 226.8148 97 4.797 248.8898 98 4.898 272.4537 99 4.999 297.569distribution of Z Normal distribution

1.48672E-06 0.00025814 2.45094E-06 0.000383666 3.9995E-06 0.000562569 6.46025E-06 0.000813598 1.03291E-05 0.001160187 1.63472E-05 0.001630742 2.56093E-05 0.002258474 3.97118E-05 0.003080516 6.09553E-05 0.004136028 9.26132E-05 0.005462942 0.000139285 0.007093014 0.000207351 0.009044929 0.000305546 0.011315317 0.000445674 0.0138678 0.000643469 0.016620542 0.000919619 0.0194332 0.001300942 0.022094744 0.001821704 0.02431417 0.002525035 0.025716663 0.003464389 0.025848163 0.004704958 0.024191389 0.006324913 0.02019609 0.008416337 0.013325453 0.011085658 0.003119197 0.014453386 -0.010728213 0.018652949 -0.028278328 0.023828414 -0.049255451 0.030130931 -0.072959218 0.03771375 -0.098195275 0.046725789 -0.123237716 0.057303789 -0.145836905 0.069563239 -0.163284321 0.083588399 -0.172542023 0.099421884 -0.170438144 0.117054396 -0.153922026 0.136415346 -0.120363821 0.157365126 -0.067874755 0.179689839 0.004383042 0.203099244 0.095932777 0.227228529 0.204801723 

# 

#  

2 4 2 

## 336 6 24 z 

## z z   

    

     

#   2 1 3 

### ( ) 1 3 ( )

6 24 z G 

### f z z z z             

## ( ) z 0.251644341 0.327454986 0.275855243 0.458868267 0.299326439 0.592750629 0.321498297 0.72191232 0.341807853 0.838755395 0.359712192 0.93584856 0.37471238 1.006534541 0.386376486 1.045510485 0.394360216 1.049320814 0.398423793 1.016708275 0.398443914 0.948782077 0.394419966 0.848980757 0.386474058 0.722829327 0.374844865 0.577512693 0.359875719 0.421307289 0.34199778 0.262927915 0.32170943 0.110854823 0.299553264 -0.027293533 0.276092166 -0.145283984 0.25188591 -0.238643935 0.227469632 -0.304862135 0.203335281 -0.343408575 0.17991684 -0.355586415 0.157579839 -0.344245108 0.136615273 -0.31339536 0.117237788 -0.267772357 0.099587708 -0.212393663 0.083736272 -0.152153024 0.069693339 -0.091482269 0.05741676 -0.034102101 0.046822635 0.017129395 0.037795734 0.060272801 0.030199481 0.094279357 0.023885038 0.118896489 0.018699163 0.134527832 0.014490659 0.142069051 0.011115369 0.142738913 0.008439746 0.137921435 0.006343145 0.129030453 0.004718997 0.117403303 0.003475077 0.104225924 0.002533081 0.090488222 0.001827693 0.076965905 0.001305351 0.064223535 0.000922829 0.052632887 0.00064578 0.042400898 0.00044732 0.0336021920.000306705 0.026212124 0.000208159 0.020137481 0.000139842 0.01524298 9.29928E-05 0.011372694 6.12113E-05 0.008366251 3.98826E-05 0.006070157 2.5722E-05 0.004344929 1.64209E-05 0.003068875 1.03767E-05 0.002139333 6.49066E-06 0.001472177 4.01874E-06 0.001000222 2.46298E-06 0.000671047 1.49417E-06 0.000444619Absolute values

0.00025814 1.30361E-05 0.000383666 3.87503E-05 0.000562569 5.68195E-05 0.000813598 8.21734E-05 0.001160187 0.000117179 0.001630742 0.000164705 0.002258474 0.000228106 0.003080516 0.000311132 0.004136028 0.000417739 0.005462942 0.000551757 0.007093014 0.000716394 0.009044929 0.000913538 0.011315317 0.001142847 0.0138678 0.001400648 0.016620542 0.001678675 0.0194332 0.001962753 0.022094744 0.002231569 0.02431417 0.002455731 0.025716663 0.002597383 0.025848163 0.002610664 0.024191389 0.00244333 0.02019609 0.002039805 0.013325453 0.001345871 0.003119197 0.000315039 0.010728213 0.00108355 0.028278328 0.002856111 0.049255451 0.004974801 0.072959218 0.007368881 0.098195275 0.009917723 0.123237716 0.012447009 0.145836905 0.014729527 0.163284321 0.016491716 0.172542023 0.017426744 0.170438144 0.017214253 0.153922026 0.015546125 0.120363821 0.012156746 0.067874755 0.00685535 0.004383042 0.000442687 0.095932777 0.009689211 0.204801723 0.020684974 

## 

#  

2 4 2 

### 336 ( ) 1 3 ( )

6 24 z 

### z z f z z z z    

       1 

## ( ) 2 () () 2 M i

i 

## xf a f x f b

 

##  

      

# 0.327454986 0.033072954 0.458868267 0.046345695 0.592750629 0.059867814 0.72191232 0.072913144 0.838755395 0.084714295 0.93584856 0.094520705 1.006534541 0.101659989 1.045510485 0.105596559 1.049320814 0.105981402 1.016708275 0.102687536 0.948782077 0.09582699 0.848980757 0.085747056 0.722829327 0.073005762 0.577512693 0.058328782 0.421307289 0.042552036 0.262927915 0.026555719 0.110854823 0.011196337 0.027293533 0.002756647 0.145283984 0.014673682 0.238643935 0.024103037 0.304862135 0.030791076 0.343408575 0.034684266 0.355586415 0.035914228 0.344245108 0.034768756 0.31339536 0.031652931 0.267772357 0.027045008 0.212393663 0.02145176 0.152153024 0.015367455 0.091482269 0.009239709 0.034102101 0.003444312 0.017129395 0.001730069 0.060272801 0.006087553 0.094279357 0.009522215 0.118896489 0.012008545 0.134527832 0.013587311 0.142069051 0.014348974 0.142738913 0.01441663 0.137921435 0.013930065 0.129030453 0.013032076 0.117403303 0.011857734 0.104225924 0.010526818 0.090488222 0.00913931 0.076965905 0.007773556 0.064223535 0.006486577 0.052632887 0.005315922 0.042400898 0.004282491 0.033602192 0.0033938210.026212124 0.002647425 0.020137481 0.002033886 0.01524298 0.001539541 0.011372694 0.001148642 0.008366251 0.000844991 0.006070157 0.000613086 0.004344929 0.000438838 0.003068875 0.000309956 0.002139333 0.000216073 0.001472177 0.00014869 0.001000222 0.000101022 0.000671047 6.77757E-05 0.000444619 2.24533E-05Cumulative normalization
1.30361E-05 7.04745E-06 5.17864E-05 2.79963E-05 0.000108606 5.87135E-05 0.000190779 0.000103137 0.000307958 0.000166486 0.000472663 0.000255527 0.000700769 0.000378843 0.001011901 0.000547045 0.00142964 0.000772879 0.001981397 0.001071165 0.002697791 0.001458456 0.003611329 0.001952325 0.004754176 0.00257016 0.006154824 0.003327366 0.007833499 0.004234876 0.009796252 0.005295962 0.012027821 0.006502373 0.014483552 0.007829968 0.017080935 0.009234142 0.0196916 0.010645496 0.02213493 0.011966388 0.024174735 0.013069129 0.025520606 0.013796721 0.025835645 0.013967035 0.026919195 0.014552814 0.029775306 0.016096859 0.034750106 0.018786291 0.042118987 0.02276999 0.05203671 0.02813162 0.064483719 0.034860611 0.079213247 0.042823556 0.095704963 0.05173916 0.113131707 0.06116025 0.13034596 0.070466465 0.145892085 0.078870872 0.15804883 0.085442943 0.164904181 0.089149021 0.165346868 0.089388343 0.175036079 0.094626437 0.195721052 0.105808962 

## ( ) 2 () () f a f x f b        0.228794006 0.123688566 0.275139701 0.14874356 0.335007515 0.181108761 0.407920659 0.220526412 0.492634954 0.266323895 0.587155658 0.317422832 0.688815647 0.372381344 0.794412206 0.42946801 0.900393608 0.486762726 1.003081144 0.542276742 1.098908134 0.594081871 1.18465519 0.640437676 1.257660952 0.679905397 1.315989734 0.711438581 1.35854177 0.734442682 1.38509749 0.748798997 1.396293827 0.75485186 1.399050474 0.756342134 1.413724156 0.764274889 1.437827193 0.777305257 1.468618269 0.793951252 1.503302535 0.812701949 1.539216763 0.832117578 1.573985519 0.850913951 1.60563845 0.868025875 1.632683458 0.882646705 1.654135218 0.894243763 1.669502674 0.90255158 1.678742383 0.907546668 1.682186695 0.909408702 1.683916764 0.910343996 1.690004317 0.913634994 1.699526532 0.918782809 1.711535077 0.925274761 1.725122388 0.932620211 1.739471362 0.940377425 1.753887993 0.948171215 1.767818058 0.955701962 1.780850133 0.962747246 1.792707867 0.969157667 1.803234685 0.974848582 1.812373996 0.979789394 1.820147552 0.983991864 1.826634129 0.987498579 1.831950051 0.990372425 1.836232541 0.992687587 1.839626363 0.9945223241.842273787 0.995953551 1.844307673 0.997053092 1.845847214 0.997885385 1.846995856 0.998506354 1.847840847 0.998963165 1.848453933 0.999294606 1.848892771 0.999531847 1.849202727 0.999699413 1.8494188 0.999816224 1.84956749 0.999896607 1.849668512 0.999951221 1.849736288 0.999987862 1.849758741 1 90.98161566**Raw Data** Runs 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 n for each source 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Sample size 0 KURTOSIS CALCULATIONS FOR SAMPLE SETS WITH n > 3 ONLY Testing Normality Raw Data Logtransformed Sample Size 0.00 0.00 Kurtosis #DIV/0! #DIV/0! http://www.spcforexcel.com/are-skewness-and-kurtosis-useful-statistics SE Kurtosis 0.00 0.00 note: cannot use this equation when n=3 because the denominator in second component is not defined (the term n-3 in denominator of second component is zero). Result Kurtosis #DIV/0! #DIV/0! Skewness #DIV/0! #DIV/0! SE Skewness 0.00 0.00 Result Raw Data Result Log Data Result Skewness #DIV/0! #DIV/0! #DIV/0! #DIV/0!
KURTOSIS CALCULATIONS FOR SAMPLE SETS WITH n = 3 ONLY Average #DIV/0! Standard Deviation #DIV/0! http://www.spiderfinancial.com/support/documentation/numxl/reference-manual/statistical-tests/test_xkurt

1 2 34 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 Testing Normality Raw Data Sample Size 0.00 Kurtosis -3.00 SE Kurtosis #DIV/0! Result Kurtosis #DIV/0! Skewness #DIV/0! SE Skewness 0.00 Result Raw Data Result Skewness #DIV/0! #DIV/0! ### The kurtosis and skewness are pulled into "Calculations for Template skew", in cells H12 and H14 respectively kurtosis= #DIV/0! depending on sample size skewness= #DIV/0! same for n>3 and n=3 ni = number test runs = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Total Number of sources = 0
n =Total # test runs = 0

1 N i df n       

###  1

N i in

  N

1 N i in

 -1

This step is for QC only:
means #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

Mean = #DIV/0!

Pooled Variance = #DIV/0!

m= number future runs = 3
Term1 #DIV/0!

Term2 #DIV/0!

Squared root Term2 #DIV/0! NOTE: the pvalue for the t-statistic is calculated as: 2*alpha, where 1-alpha is desired confidence, so if 99% confidence is desired then alpha=0.01 and 2*alpha=2*(0.01)
t-statistic = quantile t-distribution with df degrees of freedom at .99 confidence level = #NUM! UPL based on t-statistic with df at 99% confidence level #DIV/0!

0 #NUM!

#NUM! Adjustment for non-normality (skewness and kurtosis) for any sample size in particular for small samples a) n = number of test-runs = b) Calculate u0

Current Confidence Level is 2 1 0 1 1 u t n 



###  2 2 1 11n i i s xX n   



###  1 1

1 i n N ij i j X x n   

### 

1 1 n m 

21 1 s n m        1 1 i i df n

       

### 

df t 21 1

s n m        1 1 i n i ik

k i x x n  

### Conclusion interval of differences for t-values used

Adjusted UPL based on recalculated t-statistic #DIV/0!

new t-statistic from tab Recalculate t-statistic #NUM!Logtransformed

0 0 0 0 0 0 0 0 0 0 0 0 0 0

note: cannot use this equation when n=3 because the denominator in second component is not defined (the term n-3 in denominator of second component is zero).0 0 0 0 0 0 0 0 0 0 0 0 0 0#DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!Incomplete Beta Function approximation

where Inputs kurtosis= #DIV/0! x=u0= #NUM! skewness= #DIV/0!
n= 0 t0=t= #NUM!

a= -0.5 a= b= 0.5 b=
w= #NUM! w=

c.1.1. #NUM! d.1.1. c.1.2. #NUM! d.1.2. c.1.3 #NUM! d.1.3
c.1.4. #NUM! d.1.4. c.1.5. -1.5625 d.1.5.

c.1.6 #NUM! d.1.6
c.1.7 #NUM! d.1.7 Calculations Calculations Calculate the Incomplete Beta Functions needed for adjustment

c.1.) Calculate d.1.) Calculate 

### 1x w b x    

   

## , x 

### I ab a a w a a

0 

### 1 1 , 2 2 u 

### n I       



## ( ,) ( ) a  ( ) w a e w a  1 2 a w b  

3 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 1 2 6 3 w a a        

2

### 3 1 2 6 w a  

     

## ( ,)   ( ) w a e w  
a w  Result1: #NUM! Result2:

1- Calculate #NUM!

2- Calculate

2.1. Calculate #DIV/0!

2.2. Calculate #NUM!

2.3. Calculate #NUM!

Result: #DIV/0!

3-Calculate

3.1. Calculate #DIV/0!

3.2. Calculate #DIV/0!

Result #DIV/0! 0 0 

### 1 11 ( ) , 2 2 2 u 

### n P t I 

      

3 0 0 

### 2 1 1 1 1 ( ) ,1 ,1

2 2 6 2 3 2

u u 

### n n n n P t I I

n n  

###  

         

        2 2 01 2 2 0 

### 2 1 1 1 1 ( ) 6 2 1 1 n



### n t n P t n t

n  

###  



###      

     

  

   

## 2 1 6 2 nn 

2

0 

## 2 1 1 1 n t n 

 1 2 2 0 

## 1 1 n 

## t n  

##     

  

1 3 2 nn  

## 2 1 6 2 nn  

#      

1 2 4 1 1 11 11 31 ( ) , , ,

n n n n n n n n P t I I I    
                  4-Calculate

4.1. Calculate #NUM!

4.2. Calculate #DIV/0!

4.3. Calculate #DIV/0!

Result: #NUM!
5- Calculate

#NUM! 5.1. Calculate #DIV/0! 5.2. Calculate

5.3. Calculate #DIV/0!

5.4. Calculate #DIV/0!

Result #NUM! Corrected probability is:: 

#      
4 0 0 0 

# 1 2 4 1 1 11 11 31 ( ) , , , 24 2 2 12 2 2 24 2 2 u u u



# n n n n n n n n P t I I I n n  

#    
                        

0 

## 1 11

, 24 2 2 u 

## n n I         

#    0 

## 1 2 1 1

, 12 2 2

u 

## n n n

I n   

      

#    0 

### 4 1 3 1

, 24 2 2

u 

### n n n

I n   

      

#    

#    

#    

#   2 3/2 0 0

2 2

0 0



### 1 2 5 8 1 (2 5) 1 1 1 1 ( ) , ,

72 2 2 24 2 2 1 2 5 12 12 5 12 3 1 5 1 , , 24 2 2 72 2 2 u u

u u



### n n n n n n n P t I I n n n n n n n n n I I n n  

###   

   

      

                   

         

#   0 

## 1 (2 5) 1 1

, 72 2 2

u 

## n n n

I  


      

#  

#  

2 0 

## 1 2 5 8 1 1 , 24 2 2 u 

## n n n n

I n    

      

#  

#  

2 0 

## 1 2 5 12 3 1 , 24 2 2 u 

## n n n n

I n           

#  

#  

2 0 

## 1 2 5 12 5 1 , 72 2 2 u 

## n n n n

I n          

3 4 3/2 2 0 3 

### ( ) ( ) ( ) ( ) P t SkewnessP t KurtosisP t P t    

###       0.5 a= 1.5

0.5 b= 0.5 #NUM! w= #NUM!

#NUM! e.1.1. #NUM! 1.772453851 e.1.2. 0.886226925 #NUM! e.1.3 #NUM!
#NUM! e.1.4. #NUM! 0.0625 e.1.5.

#NUM! e.1.6
#NUM! e.1.7 Calculations Calculations d.1.) Calculate e.1.) Calculate 

##   3 2 2 2

2 3 

### ( ,) 1 1 5 3 1 3 11 1 , ( ) ( ) 2 2 3 2 326 3 2 3 1 1 2 6 2 w a 

### a w e w a w a I ab a a w a a

a a b b

w a w  

###    



           

                        

0 1 1 , 2 2 u n I       



## ( ,) ( ) a  ( ) w a e w a  1 2 a w b  

3 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 1 2 6 3 w a a        

2

### 3 1 2 6 w a  

     

## ( ,) ( ) a  ( ) w a e w a  1 2 a w b  

3 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 1 2 6 3 w a a        

2

### 3 1 2 6 w a  

    #NUM! Result3:



# 1 2 4 1 1 11 11 31 ( ) , , , n n n n P t I I I    
               #NUM! 0 0 0



# 1 2 4 1 1 11 11 31 ( ) , , , 24 2 2 12 2 2 24 2 2 u u u 

# n n n n P t I I I    
                     

1 1 1 1 72 2 2 24 2 2

3 1 5 1 24 2 2 72 2 2

      

                   a= 1.5 b= 0.5
w= #NUM!

f.1.1. #NUM! f.1.2. 0.886226925 f.1.3 #NUM!
f.1.4. #NUM! -0.14583 f.1.5. -0.14583

#NUM! f.1.6 #NUM!
#NUM! f.1.7 #NUM!

Calculations f.1.) Calculate 0 3 1 , 2 2 u n I        0 5 1 , 2 2 u n I       



## ( ,) ( ) a  ( ) w a e w a  1 2 a w b  

3 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 1 2 6 3 w a a        

2

### 3 1 2 6 w a  

    #NUM! Result4: #NUM!a= -0.5 a= 0.5

b= 1 b= 1
w= #NUM! w= #NUM!
g.1.1. #NUM! h.1.1. g.1.2. #NUM! h.1.2. g.1.3 #NUM! h.1.3 g.1.4. #NUM! h.1.4. g.1.5. -1.5625 h.1.5.
g.1.6 #NUM! h.1.6 g.1.7 #NUM! h.1.7 h.1.) Calculate

Calculations Calculations g.1.) Calculate 0 1,1 2 u n I       



## ( ,) ( ) a  ( ) w a e w a  1 2 a w b  

3 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 1 2 6 3 w a a        

2

### 3 1 2 6 w a  

     

## ( ,) ( ) a  ( ) w a e w a  1 2 a w b  

3 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 1 2 6 3 w a a        

2

### 3 1 2 6 w a  

    Result5: #NUM! Result6:#NUM!

1.772454 #NUM!
#NUM! 0.0625

#NUM!
#NUM! 0 1,1 2 u n I       

5 3 1 2 3 2 3 a a   

2 

### 3 11 1 2 6 3 w a a         3 1 2 6 w a  

    #NUM!#NUM!

#NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! 4) If you don't get a YES in row 11, play around with the cell G3, the interval for differences until you get one p-value closed to 0.01 5) Once you get a YES, use the t-statistic in the calculations of the UPL. Go back to Template_nonnormal tab and paste the value in row 5 that has a Yes in row 11. Copy the new t-value in cell E164 of**Template_nonnormal n<3** 1) Start with initial t-statistic from Template 2) Determine interval of differences for t-values 2) Candidate values for t-statistic 3) Corrected p-values Acutal significance level with t-value 4) Is p-value closed enough to 0.01?t-statistic=

Incomplete Beta Function approximation

where Inputs kurtosis= #DIV/0! x=u0= #NUM! skewness= #DIV/0!
n= 0 t0=t= #NUM!

a= -0.5 a= b= 0.5 b=
w= #NUM! w=

c.1.1. #NUM! d.1.1. c.1.2. #NUM! d.1.2. c.1.3 #NUM! d.1.3
c.1.4. #NUM! d.1.4. c.1.5. -1.5625 d.1.5.

c.1.6 #NUM! d.1.6 Calculations Calculations Calculate u0. Sustitute each candidate t-value in u0 equation

Calculate the Incomplete Beta Functions needed for adjustment

c.1.) Calculate d.1.) Calculate 

### 1x w b x    

   

##   , x 

### I ab a a w a a         

0 

### 1 1 , 2 2 u 

### n I       



## ( ,) ( ) a  ( ) w a e w a  1 2 a w b  

3 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 1 2 6 3 w a a         

## ( ,) ( ) a  ( ) w a e w a  1 2 a w b   2 

### 1 0 1 1 u t n 

c.1.7 #NUM! d.1.7

Result1: #NUM! Result2:
1- Calculate #NUM!

2- Calculate

2.1. Calculate #DIV/0!

2.2. Calculate #NUM!

2.3. Calculate #NUM!

Result: #DIV/0!

3-Calculate

3.1. Calculate #DIV/0!

3.2. Calculate #DIV/0!



### 2 6 3    

2

### 3 1 2 6 w a  

    

0 0 

### 1 11 ( ) , 2 2 2 u 

### n P t I 

      

3 0 0 

### 2 1 1 1 1 ( ) ,1 ,1

2 2 6 2 3 2

u u 

### n n n n P t I I

n n  

###  

         

        2 2 01 2 2 0 

### 2 1 1 1 1 ( ) 6 2 1 1 n



### n t n P t n t

n  

###  



###      

     

  

   

## 2 1 6 2 nn 

2

0 

## 2 1 1 1 n t n 

 1 2 2 0 

## 1 1 n 

## t n  

##     

  

1 3 2 nn  

## 2 1 6 2 nn Result #DIV/0!

4-Calculate
4.1. Calculate #NUM!

4.2. Calculate #DIV/0!

4.3. Calculate #DIV/0!

Result: #NUM!
5- Calculate

#NUM! 5.1. Calculate #DIV/0! 5.2. Calculate

5.3. Calculate #DIV/0!

5.4. Calculate #DIV/0!

Result #NUM! 

#      
4 0 0 0 

# 1 2 4 1 1 11 11 31 ( ) , , , 24 2 2 12 2 2 24 2 2 u u u



# n n n n n n n n P t I I I n n  

#    
                        

0 

## 1 11

, 24 2 2 u 

## n n I         

#    0 

## 1 2 1 1

, 12 2 2

u 

## n n n

I n   

      

#    0 

### 4 1 3 1

, 24 2 2

u 

### n n n

I n   

      

#    

#    

#    

#   2 3/2 0 0

2 2

0 0



### 1 2 5 8 1 (2 5) 1 1 1 1 ( ) , ,

72 2 2 24 2 2 1 2 5 12 12 5 12 3 1 5 1 , , 24 2 2 72 2 2 u u

u u



### n n n n n n n P t I I n n n n n n n n n I I n n  

###   

   

      

                   

         

#   0 

## 1 (2 5) 1 1

, 72 2 2

u 

## n n n

I  


      

#  

#  

2 0 

## 1 2 5 8 1 1 , 24 2 2 u 

## n n n n

I n    

      

#  

#  

2 0 

## 1 2 5 12 3 1 , 24 2 2 u 

## n n n n

I n           

#  

#  

2 0 

## 1 2 5 12 5 1 , 72 2 2 u 

## n n n n

I n          Corrected probability is:: 3 4 3/2 2 0 3 

### ( ) ( ) ( ) ( ) P t SkewnessP t KurtosisP t P t    

###       #NUM! u0= #NUM!

0.5 a= 1.5 0.5 b= 0.5 #NUM! w= #NUM!

#NUM! e.1.1. #NUM! 1.772453851 e.1.2. 0.886226925 #NUM! e.1.3 #NUM!
#NUM! e.1.4. #NUM! 0.0625 e.1.5.

#NUM! e.1.6 Calculations e.1.) Calculate 

##  3 2 2 2

2 3 

### ( ,) 1 1 5 3 1 3 11 1 , ( ) ( ) 2 2 3 2 326 3 2 3 1 1 2 6 2 w a 

### a w e w a w a I ab a a w a a

a a b b

w a w  

###    



           

                        

0 1 1 , 2 2 u n I       



## ( ,) ( ) a  ( ) w a e w a 1 2 a w b  

3 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 1 2 6 3 w a a         

## ( ,) ( ) a  ( ) w a e w a  1 2 a w b  

3 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 1 2 6 3 w a a        #NUM! e.1.7

#NUM! Result3: 

### 2 6 3    

2

### 3 1 2 6 w a  

     

### 2 6 3    

2

### 3 1 2 6 w a  

    0 0 0 

# 1 2 4 1 1 11 11 31 ( ) , , , 24 2 2 12 2 2 24 2 2 u u u 

# n n n n P t I I I                      #NUM!a= 1.5 b= 0.5
w= #NUM!

f.1.1. #NUM! f.1.2. 0.886226925 f.1.3 #NUM!
f.1.4. #NUM! -0.14583 f.1.5. -0.14583

#NUM! f.1.6 #NUM!

Calculations f.1.) Calculate 0 3 1 , 2 2 u n I        0 5 1 , 2 2 u n I       



## ( ,) ( ) a  ( ) w a e w a  1 2 a w b  

3 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 1 2 6 3 w a a        #NUM! f.1.7 #NUM!

#NUM! Result4: #NUM! 

### 2 6 3    

2

### 3 1 2 6 w a  

    a= -0.5 a= 0.5

b= 1 b= 1
w= #NUM! w= #NUM!
g.1.1. #NUM! h.1.1. g.1.2. #NUM! h.1.2. g.1.3 #NUM! h.1.3 g.1.4. #NUM! h.1.4. g.1.5. -1.5625 h.1.5.
g.1.6 #NUM! h.1.6 g.1.) Calculate h.1.) Calculate

Calculations Calculations 0 1,1 2 u n I       



## ( ,) ( ) a  ( ) w a e w a  1 2 a w b  

3 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 1 2 6 3 w a a         

## ( ,) ( ) a  ( ) w a e w a  1 2 a w b  

3 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 1 2 6 3 w a a        g.1.7 #NUM! h.1.7

Result5: #NUM! Result6:



### 2 6 3    

2

### 3 1 2 6 w a  

     

### 2 6 3    

2

### 3 1 2 6 w a  

    #NUM!

1.772454 #NUM!
#NUM! 0.0625

#NUM! 0 1,1 2 u n I       

5 3 1 2 3 2 3 a a   

2 

### 3 11 1 2 6 3 w a a        #NUM!

#NUM! 

### 2 6 3     3 1 2 6 w a  

    t-statistic=

Incomplete Beta Function approximation

where Inputs kurtosis= #DIV/0! x=u0= #NUM! skewness= #DIV/0!
n= 0 t0=t= #NUM!

a= -0.5 a= b= 0.5 b=
w= #NUM! w=

c.1.1. #NUM! d.1.1. c.1.2. #NUM! d.1.2. c.1.3 #NUM! d.1.3
c.1.4. #NUM! d.1.4. c.1.5. -1.5625 d.1.5.

c.1.6 #NUM! d.1.6 Calculations Calculations Calculate u0. Sustitute each candidate t-value in u0 equation

Calculate the Incomplete Beta Functions needed for adjustment

c.1.) Calculate d.1.) Calculate 

### 1x w b x    

   

##   , x 

### I ab a a w a a         

0 

### 1 1 , 2 2 u 

### n I       



## ( ,) ( ) a  ( ) w a e w a  1 2 a w b  

3 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 1 2 6 3 w a a         

## ( ,) ( ) a  ( ) w a e w a  1 2 a w b   2 

### 1 0 1 1 u t n 

c.1.7 #NUM! d.1.7

Result1: #NUM! Result2:
1- Calculate #NUM!

2- Calculate

2.1. Calculate #DIV/0!

2.2. Calculate #NUM!

2.3. Calculate #NUM!

Result: #DIV/0!

3-Calculate

3.1. Calculate #DIV/0!

3.2. Calculate #DIV/0!



### 2 6 3    

2

### 3 1 2 6 w a  

    

0 0 

### 1 11 ( ) , 2 2 2 u 

### n P t I 

      

3 0 0 

### 2 1 1 1 1 ( ) ,1 ,1

2 2 6 2 3 2

u u 

### n n n n P t I I

n n  

###  

         

        2 2 01 2 2 0 

### 2 1 1 1 1 ( ) 6 2 1 1 n



### n t n P t n t

n  

###  



###      

     

  

   

## 2 1 6 2 nn 

2

0 

## 2 1 1 1 n t n 

 1 2 2 0 

## 1 1 n 

## t n  

##     

  

1 3 2 nn  

## 2 1 6 2 nn Result #DIV/0!

4-Calculate
4.1. Calculate #NUM!

4.2. Calculate #DIV/0!

4.3. Calculate #DIV/0!

Result: #NUM!
5- Calculate

#NUM! 5.1. Calculate #DIV/0! 5.2. Calculate

5.3. Calculate #DIV/0!

5.4. Calculate #DIV/0!

Result #NUM! 

#      
4 0 0 0 

# 1 2 4 1 1 11 11 31 ( ) , , , 24 2 2 12 2 2 24 2 2 u u u



# n n n n n n n n P t I I I n n  

#    
                        

0 

## 1 11

, 24 2 2 u 

## n n I         

#    0 

## 1 2 1 1

, 12 2 2

u 

## n n n

I n   

      

#    0 

### 4 1 3 1

, 24 2 2

u 

### n n n

I n   

      

#    

#    

#    

#   2 3/2 0 0

2 2

0 0



### 1 2 5 8 1 (2 5) 1 1 1 1 ( ) , ,

72 2 2 24 2 2 1 2 5 12 12 5 12 3 1 5 1 , , 24 2 2 72 2 2 u u

u u



### n n n n n n n P t I I n n n n n n n n n I I n n  

###   

   

      

                   

         

#   0 

## 1 (2 5) 1 1

, 72 2 2

u 

## n n n

I  


      

#  

#  

2 0 

## 1 2 5 8 1 1 , 24 2 2 u 

## n n n n

I n    

      

#  

#  

2 0 

## 1 2 5 12 3 1 , 24 2 2 u 

## n n n n

I n           

#  

#  

2 0 

## 1 2 5 12 5 1 , 72 2 2 u 

## n n n n

I n          Corrected probability is:: 3 4 3/2 2 0 3 

### ( ) ( ) ( ) ( ) P t SkewnessP t KurtosisP t P t    

###       #NUM! u0= #NUM!

0.5 a= 1.5 0.5 b= 0.5 #NUM! w= #NUM!

#NUM! e.1.1. #NUM! 1.772453851 e.1.2. 0.886226925 #NUM! e.1.3 #NUM!
#NUM! e.1.4. #NUM! 0.0625 e.1.5.

#NUM! e.1.6 Calculations e.1.) Calculate 

##  3 2 2 2

2 3 

### ( ,) 1 1 5 3 1 3 11 1 , ( ) ( ) 2 2 3 2 326 3 2 3 1 1 2 6 2 w a 

### a w e w a w a I ab a a w a a

a a b b

w a w  

###    



           

                        

0 1 1 , 2 2 u n I       



## ( ,) ( ) a  ( ) w a e w a 1 2 a w b  

3 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 1 2 6 3 w a a         

## ( ,) ( ) a  ( ) w a e w a  1 2 a w b  

3 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 1 2 6 3 w a a        #NUM! e.1.7

#NUM! Result3: 

### 2 6 3    

2

### 3 1 2 6 w a  

     

### 2 6 3    

2

### 3 1 2 6 w a  

    0 0 0 

# 1 2 4 1 1 11 11 31 ( ) , , , 24 2 2 12 2 2 24 2 2 u u u 

# n n n n P t I I I                      #NUM!a= 1.5 b= 0.5
w= #NUM!

f.1.1. #NUM! f.1.2. 0.886226925 f.1.3 #NUM!
f.1.4. #NUM! -0.14583 f.1.5. -0.14583

#NUM! f.1.6 #NUM!

Calculations f.1.) Calculate 0 3 1 , 2 2 u n I        0 5 1 , 2 2 u n I       



## ( ,) ( ) a  ( ) w a e w a  1 2 a w b  

3 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 1 2 6 3 w a a        #NUM! f.1.7 #NUM!

#NUM! Result4: #NUM! 

### 2 6 3    

2

### 3 1 2 6 w a  

    a= -0.5 a= 0.5

b= 1 b= 1
w= #NUM! w= #NUM!
g.1.1. #NUM! h.1.1. g.1.2. #NUM! h.1.2. g.1.3 #NUM! h.1.3 g.1.4. #NUM! h.1.4. g.1.5. -1.5625 h.1.5.
g.1.6 #NUM! h.1.6 g.1.) Calculate h.1.) Calculate

Calculations Calculations 0 1,1 2 u n I       



## ( ,) ( ) a  ( ) w a e w a  1 2 a w b  

3 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 1 2 6 3 w a a         

## ( ,) ( ) a  ( ) w a e w a  1 2 a w b  

3 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 1 2 6 3 w a a        g.1.7 #NUM! h.1.7

Result5: #NUM! Result6:



### 2 6 3    

2

### 3 1 2 6 w a  

     

### 2 6 3    

2

### 3 1 2 6 w a  

    #NUM!

1.772454 #NUM!
#NUM! 0.0625

#NUM! 0 1,1 2 u n I       

5 3 1 2 3 2 3 a a   

2 

### 3 11 1 2 6 3 w a a        #NUM!

#NUM! 

### 2 6 3     3 1 2 6 w a  

    t-statistic=

Incomplete Beta Function approximation

where Inputs kurtosis= #DIV/0! x=u0= #NUM! skewness= #DIV/0!
n= 0 t0=t= #NUM!

a= -0.5 a= b= 0.5 b=
w= #NUM! w=

c.1.1. #NUM! d.1.1. c.1.2. #NUM! d.1.2. c.1.3 #NUM! d.1.3
c.1.4. #NUM! d.1.4. c.1.5. -1.5625 d.1.5.

c.1.6 #NUM! d.1.6 Calculations Calculations Calculate u0. Sustitute each candidate t-value in u0 equation

Calculate the Incomplete Beta Functions needed for adjustment

c.1.) Calculate d.1.) Calculate 

### 1x w b x    

   

##   , x 

### I ab a a w a a         

0 

### 1 1 , 2 2 u 

### n I       



## ( ,) ( ) a  ( ) w a e w a  1 2 a w b  

3 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 1 2 6 3 w a a         

## ( ,) ( ) a  ( ) w a e w a  1 2 a w b   2 

### 1 0 1 1 u t n 

c.1.7 #NUM! d.1.7

Result1: #NUM! Result2:
1- Calculate #NUM!

2- Calculate

2.1. Calculate #DIV/0!

2.2. Calculate #NUM!

2.3. Calculate #NUM!

Result: #DIV/0!

3-Calculate

3.1. Calculate #DIV/0!

3.2. Calculate #DIV/0!



### 2 6 3    

2

### 3 1 2 6 w a  

    

0 0 

### 1 11 ( ) , 2 2 2 u 

### n P t I 

      

3 0 0 

### 2 1 1 1 1 ( ) ,1 ,1

2 2 6 2 3 2

u u 

### n n n n P t I I

n n  

###  

         

        2 2 01 2 2 0 

### 2 1 1 1 1 ( ) 6 2 1 1 n



### n t n P t n t

n  

###  



###      

     

  

   

## 2 1 6 2 nn 

2

0 

## 2 1 1 1 n t n 

 1 2 2 0 

## 1 1 n 

## t n  

##     

  

1 3 2 nn  

## 2 1 6 2 nn Result #DIV/0!

4-Calculate
4.1. Calculate #NUM!

4.2. Calculate #DIV/0!

4.3. Calculate #DIV/0!

Result: #NUM!
5- Calculate

#NUM! 5.1. Calculate #DIV/0! 5.2. Calculate

5.3. Calculate #DIV/0!

5.4. Calculate #DIV/0!

Result #NUM! 

#      
4 0 0 0 

# 1 2 4 1 1 11 11 31 ( ) , , , 24 2 2 12 2 2 24 2 2 u u u



# n n n n n n n n P t I I I n n  

#    
                        

0 

## 1 11

, 24 2 2 u 

## n n I         

#    0 

## 1 2 1 1

, 12 2 2

u 

## n n n

I n   

      

#    0 

### 4 1 3 1

, 24 2 2

u 

### n n n

I n   

      

#    

#    

#    

#   2 3/2 0 0

2 2

0 0



### 1 2 5 8 1 (2 5) 1 1 1 1 ( ) , ,

72 2 2 24 2 2 1 2 5 12 12 5 12 3 1 5 1 , , 24 2 2 72 2 2 u u

u u



### n n n n n n n P t I I n n n n n n n n n I I n n  

###   

   

      

                   

         

#   0 

## 1 (2 5) 1 1

, 72 2 2

u 

## n n n

I  


      

#  

#  

2 0 

## 1 2 5 8 1 1 , 24 2 2 u 

## n n n n

I n    

      

#  

#  

2 0 

## 1 2 5 12 3 1 , 24 2 2 u 

## n n n n

I n           

#  

#  

2 0 

## 1 2 5 12 5 1 , 72 2 2 u 

## n n n n

I n          Corrected probability is:: 3 4 3/2 2 0 3 

### ( ) ( ) ( ) ( ) P t SkewnessP t KurtosisP t P t    

###       #NUM! u0= #NUM!

0.5 a= 1.5 0.5 b= 0.5 #NUM! w= #NUM!

#NUM! e.1.1. #NUM! 1.772453851 e.1.2. 0.886226925 #NUM! e.1.3 #NUM!
#NUM! e.1.4. #NUM! 0.0625 e.1.5.

#NUM! e.1.6 Calculations e.1.) Calculate 

##  3 2 2 2

2 3 

### ( ,) 1 1 5 3 1 3 11 1 , ( ) ( ) 2 2 3 2 326 3 2 3 1 1 2 6 2 w a 

### a w e w a w a I ab a a w a a

a a b b

w a w  

###    



           

                        

0 1 1 , 2 2 u n I       



## ( ,) ( ) a  ( ) w a e w a 1 2 a w b  

3 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 1 2 6 3 w a a         

## ( ,) ( ) a  ( ) w a e w a  1 2 a w b  

3 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 1 2 6 3 w a a        #NUM! e.1.7

#NUM! Result3: 

### 2 6 3    

2

### 3 1 2 6 w a  

     

### 2 6 3    

2

### 3 1 2 6 w a  

    0 0 0 

# 1 2 4 1 1 11 11 31 ( ) , , , 24 2 2 12 2 2 24 2 2 u u u 

# n n n n P t I I I                      #NUM!a= 1.5 b= 0.5
w= #NUM!

f.1.1. #NUM! f.1.2. 0.886226925 f.1.3 #NUM!
f.1.4. #NUM! -0.14583 f.1.5. -0.14583

#NUM! f.1.6 #NUM!

Calculations f.1.) Calculate 0 3 1 , 2 2 u n I        0 5 1 , 2 2 u n I       



## ( ,) ( ) a  ( ) w a e w a  1 2 a w b  

3 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 1 2 6 3 w a a        #NUM! f.1.7 #NUM!

#NUM! Result4: #NUM! 

### 2 6 3    

2

### 3 1 2 6 w a  

    a= -0.5 a= 0.5

b= 1 b= 1
w= #NUM! w= #NUM!
g.1.1. #NUM! h.1.1. g.1.2. #NUM! h.1.2. g.1.3 #NUM! h.1.3 g.1.4. #NUM! h.1.4. g.1.5. -1.5625 h.1.5.
g.1.6 #NUM! h.1.6 g.1.) Calculate h.1.) Calculate

Calculations Calculations 0 1,1 2 u n I       



## ( ,) ( ) a  ( ) w a e w a  1 2 a w b  

3 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 1 2 6 3 w a a         

## ( ,) ( ) a  ( ) w a e w a  1 2 a w b  

3 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 1 2 6 3 w a a        g.1.7 #NUM! h.1.7

Result5: #NUM! Result6:



### 2 6 3    

2

### 3 1 2 6 w a  

     

### 2 6 3    

2

### 3 1 2 6 w a  

    #NUM!

1.772454 #NUM!
#NUM! 0.0625

#NUM! 0 1,1 2 u n I       

5 3 1 2 3 2 3 a a   

2 

### 3 11 1 2 6 3 w a a        #NUM!

#NUM! 

### 2 6 3     3 1 2 6 w a  

    t-statistic=

Incomplete Beta Function approximation

where Inputs kurtosis= #DIV/0! x=u0= #NUM! skewness= #DIV/0!
n= 0 t0=t= #NUM!

a= -0.5 a= b= 0.5 b=
w= #NUM! w=

c.1.1. #NUM! d.1.1. c.1.2. #NUM! d.1.2. c.1.3 #NUM! d.1.3
c.1.4. #NUM! d.1.4. c.1.5. -1.5625 d.1.5.

c.1.6 #NUM! d.1.6 Calculations Calculations Calculate u0. Sustitute each candidate t-value in u0 equation

Calculate the Incomplete Beta Functions needed for adjustment

c.1.) Calculate d.1.) Calculate 

### 1x w b x    

   

##   , x 

### I ab a a w a a         

0 

### 1 1 , 2 2 u 

### n I       



## ( ,) ( ) a  ( ) w a e w a  1 2 a w b  

3 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 1 2 6 3 w a a         

## ( ,) ( ) a  ( ) w a e w a  1 2 a w b   2 

### 1 0 1 1 u t n 

c.1.7 #NUM! d.1.7

Result1: #NUM! Result2:
1- Calculate #NUM!

2- Calculate

2.1. Calculate #DIV/0!

2.2. Calculate #NUM!

2.3. Calculate #NUM!

Result: #DIV/0!

3-Calculate

3.1. Calculate #DIV/0!

3.2. Calculate #DIV/0!



### 2 6 3    

2

### 3 1 2 6 w a  

    

0 0 

### 1 11 ( ) , 2 2 2 u 

### n P t I 

      

3 0 0 

### 2 1 1 1 1 ( ) ,1 ,1

2 2 6 2 3 2

u u 

### n n n n P t I I

n n  

###  

         

        2 2 01 2 2 0 

### 2 1 1 1 1 ( ) 6 2 1 1 n



### n t n P t n t

n  

###  



###      

     

  

   

## 2 1 6 2 nn 

2

0 

## 2 1 1 1 n t n 

 1 2 2 0 

## 1 1 n 

## t n  

##     

  

1 3 2 nn  

## 2 1 6 2 nn Result #DIV/0!

4-Calculate
4.1. Calculate #NUM!

4.2. Calculate #DIV/0!

4.3. Calculate #DIV/0!

Result: #NUM!
5- Calculate

#NUM! 5.1. Calculate #DIV/0! 5.2. Calculate

5.3. Calculate #DIV/0!

5.4. Calculate #DIV/0!

Result #NUM! 

#      
4 0 0 0 

# 1 2 4 1 1 11 11 31 ( ) , , , 24 2 2 12 2 2 24 2 2 u u u



# n n n n n n n n P t I I I n n  

#    
                        

0 

## 1 11

, 24 2 2 u 

## n n I         

#    0 

## 1 2 1 1

, 12 2 2

u 

## n n n

I n   

      

#    0 

### 4 1 3 1

, 24 2 2

u 

### n n n

I n   

      

#    

#    

#    

#   2 3/2 0 0

2 2

0 0



### 1 2 5 8 1 (2 5) 1 1 1 1 ( ) , ,

72 2 2 24 2 2 1 2 5 12 12 5 12 3 1 5 1 , , 24 2 2 72 2 2 u u

u u



### n n n n n n n P t I I n n n n n n n n n I I n n  

###   

   

      

                   

         

#   0 

## 1 (2 5) 1 1

, 72 2 2

u 

## n n n

I  


      

#  

#  

2 0 

## 1 2 5 8 1 1 , 24 2 2 u 

## n n n n

I n    

      

#  

#  

2 0 

## 1 2 5 12 3 1 , 24 2 2 u 

## n n n n

I n           

#  

#  

2 0 

## 1 2 5 12 5 1 , 72 2 2 u 

## n n n n

I n          Corrected probability is:: 3 4 3/2 2 0 3 

### ( ) ( ) ( ) ( ) P t SkewnessP t KurtosisP t P t    

###       #NUM! u0= #NUM!

0.5 a= 1.5 0.5 b= 0.5 #NUM! w= #NUM!

#NUM! e.1.1. #NUM! 1.772453851 e.1.2. 0.886226925 #NUM! e.1.3 #NUM!
#NUM! e.1.4. #NUM! 0.0625 e.1.5.

#NUM! e.1.6 Calculations e.1.) Calculate 

##  3 2 2 2

2 3 

### ( ,) 1 1 5 3 1 3 11 1 , ( ) ( ) 2 2 3 2 326 3 2 3 1 1 2 6 2 w a 

### a w e w a w a I ab a a w a a

a a b b

w a w  

###    



           

                        

0 1 1 , 2 2 u n I       



## ( ,) ( ) a  ( ) w a e w a 1 2 a w b  

3 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 1 2 6 3 w a a         

## ( ,) ( ) a  ( ) w a e w a  1 2 a w b  

3 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 1 2 6 3 w a a        #NUM! e.1.7

#NUM! Result3: 

### 2 6 3    

2

### 3 1 2 6 w a  

     

### 2 6 3    

2

### 3 1 2 6 w a  

    0 0 0 

# 1 2 4 1 1 11 11 31 ( ) , , , 24 2 2 12 2 2 24 2 2 u u u 

# n n n n P t I I I                      #NUM!a= 1.5 b= 0.5
w= #NUM!

f.1.1. #NUM! f.1.2. 0.886226925 f.1.3 #NUM!
f.1.4. #NUM! -0.14583 f.1.5. -0.14583

#NUM! f.1.6 #NUM!

Calculations f.1.) Calculate 0 3 1 , 2 2 u n I        0 5 1 , 2 2 u n I       



## ( ,) ( ) a  ( ) w a e w a  1 2 a w b  

3 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 1 2 6 3 w a a        #NUM! f.1.7 #NUM!

#NUM! Result4: #NUM! 

### 2 6 3    

2

### 3 1 2 6 w a  

    a= -0.5 a= 0.5

b= 1 b= 1
w= #NUM! w= #NUM!
g.1.1. #NUM! h.1.1. g.1.2. #NUM! h.1.2. g.1.3 #NUM! h.1.3 g.1.4. #NUM! h.1.4. g.1.5. -1.5625 h.1.5.
g.1.6 #NUM! h.1.6 g.1.) Calculate h.1.) Calculate

Calculations Calculations 0 1,1 2 u n I       



## ( ,) ( ) a  ( ) w a e w a  1 2 a w b  

3 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 1 2 6 3 w a a         

## ( ,) ( ) a  ( ) w a e w a  1 2 a w b  

3 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 1 2 6 3 w a a        g.1.7 #NUM! h.1.7

Result5: #NUM! Result6:



### 2 6 3    

2

### 3 1 2 6 w a  

     

### 2 6 3    

2

### 3 1 2 6 w a  

    #NUM!

1.772454 #NUM!
#NUM! 0.0625

#NUM! 0 1,1 2 u n I       

5 3 1 2 3 2 3 a a   

2 

### 3 11 1 2 6 3 w a a        #NUM!

#NUM! 

### 2 6 3     3 1 2 6 w a  

    t-statistic=

Incomplete Beta Function approximation

where Inputs kurtosis= #DIV/0! x=u0= #NUM! skewness= #DIV/0!
n= 0 t0=t= #NUM!

a= -0.5 a= b= 0.5 b=
w= #NUM! w=

c.1.1. #NUM! d.1.1. c.1.2. #NUM! d.1.2. c.1.3 #NUM! d.1.3
c.1.4. #NUM! d.1.4. c.1.5. -1.5625 d.1.5.

c.1.6 #NUM! d.1.6 Calculations Calculations Calculate u0. Sustitute each candidate t-value in u0 equation

Calculate the Incomplete Beta Functions needed for adjustment

c.1.) Calculate d.1.) Calculate 

### 1x w b x    

   

##   , x 

### I ab a a w a a         

0 

### 1 1 , 2 2 u 

### n I       



## ( ,) ( ) a  ( ) w a e w a  1 2 a w b  

3 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 1 2 6 3 w a a         

## ( ,) ( ) a  ( ) w a e w a  1 2 a w b   2 

### 1 0 1 1 u t n 

c.1.7 #NUM! d.1.7

Result1: #NUM! Result2:
1- Calculate #NUM!

2- Calculate

2.1. Calculate #DIV/0!

2.2. Calculate #NUM!

2.3. Calculate #NUM!

Result: #DIV/0!

3-Calculate

3.1. Calculate #DIV/0!

3.2. Calculate #DIV/0!



### 2 6 3    

2

### 3 1 2 6 w a  

    

0 0 

### 1 11 ( ) , 2 2 2 u 

### n P t I 

      

3 0 0 

### 2 1 1 1 1 ( ) ,1 ,1

2 2 6 2 3 2

u u 

### n n n n P t I I

n n  

###  

         

        2 2 01 2 2 0 

### 2 1 1 1 1 ( ) 6 2 1 1 n



### n t n P t n t

n  

###  



###      

     

  

   

## 2 1 6 2 nn 

2

0 

## 2 1 1 1 n t n 

 1 2 2 0 

## 1 1 n 

## t n  

##     

  

1 3 2 nn  

## 2 1 6 2 nn Result #DIV/0!

4-Calculate
4.1. Calculate #NUM!

4.2. Calculate #DIV/0!

4.3. Calculate #DIV/0!

Result: #NUM!
5- Calculate

#NUM! 5.1. Calculate #DIV/0! 5.2. Calculate

5.3. Calculate #DIV/0!

5.4. Calculate #DIV/0!

Result #NUM! 

#      
4 0 0 0 

# 1 2 4 1 1 11 11 31 ( ) , , , 24 2 2 12 2 2 24 2 2 u u u



# n n n n n n n n P t I I I n n  

#    
                        

0 

## 1 11

, 24 2 2 u 

## n n I         

#    0 

## 1 2 1 1

, 12 2 2

u 

## n n n

I n   

      

#    0 

### 4 1 3 1

, 24 2 2

u 

### n n n

I n   

      

#    

#    

#    

#   2 3/2 0 0

2 2

0 0



### 1 2 5 8 1 (2 5) 1 1 1 1 ( ) , ,

72 2 2 24 2 2 1 2 5 12 12 5 12 3 1 5 1 , , 24 2 2 72 2 2 u u

u u



### n n n n n n n P t I I n n n n n n n n n I I n n  

###   

   

      

                   

         

#   0 

## 1 (2 5) 1 1

, 72 2 2

u 

## n n n

I  


      

#  

#  

2 0 

## 1 2 5 8 1 1 , 24 2 2 u 

## n n n n

I n    

      

#  

#  

2 0 

## 1 2 5 12 3 1 , 24 2 2 u 

## n n n n

I n           

#  

#  

2 0 

## 1 2 5 12 5 1 , 72 2 2 u 

## n n n n

I n          Corrected probability is:: 3 4 3/2 2 0 3 

### ( ) ( ) ( ) ( ) P t SkewnessP t KurtosisP t P t    

###       #NUM! u0= #NUM!

0.5 a= 1.5 0.5 b= 0.5 #NUM! w= #NUM!

#NUM! e.1.1. #NUM! 1.772453851 e.1.2. 0.886226925 #NUM! e.1.3 #NUM!
#NUM! e.1.4. #NUM! 0.0625 e.1.5.

#NUM! e.1.6 Calculations e.1.) Calculate 

##  3 2 2 2

2 3 

### ( ,) 1 1 5 3 1 3 11 1 , ( ) ( ) 2 2 3 2 326 3 2 3 1 1 2 6 2 w a 

### a w e w a w a I ab a a w a a

a a b b

w a w  

###    



           

                        

0 1 1 , 2 2 u n I       



## ( ,) ( ) a  ( ) w a e w a 1 2 a w b  

3 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 1 2 6 3 w a a         

## ( ,) ( ) a  ( ) w a e w a  1 2 a w b  

3 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 1 2 6 3 w a a        #NUM! e.1.7

#NUM! Result3: 

### 2 6 3    

2

### 3 1 2 6 w a  

     

### 2 6 3    

2

### 3 1 2 6 w a  

    0 0 0 

# 1 2 4 1 1 11 11 31 ( ) , , , 24 2 2 12 2 2 24 2 2 u u u 

# n n n n P t I I I                      #NUM!a= 1.5 b= 0.5
w= #NUM!

f.1.1. #NUM! f.1.2. 0.886226925 f.1.3 #NUM!
f.1.4. #NUM! -0.14583 f.1.5. -0.14583

#NUM! f.1.6 #NUM!

Calculations f.1.) Calculate 0 3 1 , 2 2 u n I        0 5 1 , 2 2 u n I       



## ( ,) ( ) a  ( ) w a e w a  1 2 a w b  

3 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 1 2 6 3 w a a        #NUM! f.1.7 #NUM!

#NUM! Result4: #NUM! 

### 2 6 3    

2

### 3 1 2 6 w a  

    a= -0.5 a= 0.5

b= 1 b= 1
w= #NUM! w= #NUM!
g.1.1. #NUM! h.1.1. g.1.2. #NUM! h.1.2. g.1.3 #NUM! h.1.3 g.1.4. #NUM! h.1.4. g.1.5. -1.5625 h.1.5.
g.1.6 #NUM! h.1.6 g.1.) Calculate h.1.) Calculate

Calculations Calculations 0 1,1 2 u n I       



## ( ,) ( ) a  ( ) w a e w a  1 2 a w b  

3 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 1 2 6 3 w a a         

## ( ,) ( ) a  ( ) w a e w a  1 2 a w b  

3 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 1 2 6 3 w a a        g.1.7 #NUM! h.1.7

Result5: #NUM! Result6:



### 2 6 3    

2

### 3 1 2 6 w a  

     

### 2 6 3    

2

### 3 1 2 6 w a  

    #NUM!

1.772454 #NUM!
#NUM! 0.0625

#NUM! 0 1,1 2 u n I       

5 3 1 2 3 2 3 a a   

2 

### 3 11 1 2 6 3 w a a        #NUM!

#NUM! 

### 2 6 3     3 1 2 6 w a  

    t-statistic=

Incomplete Beta Function approximation

where Inputs kurtosis= #DIV/0! x=u0= #NUM! skewness= #DIV/0!
n= 0 t0=t= #NUM!

a= -0.5 a= b= 0.5 b=
w= #NUM! w=

c.1.1. #NUM! d.1.1. c.1.2. #NUM! d.1.2. c.1.3 #NUM! d.1.3
c.1.4. #NUM! d.1.4. c.1.5. -1.5625 d.1.5.

c.1.6 #NUM! d.1.6 Calculations Calculations Calculate u0. Sustitute each candidate t-value in u0 equation

Calculate the Incomplete Beta Functions needed for adjustment

c.1.) Calculate d.1.) Calculate 

### 1x w b x    

   

##   , x 

### I ab a a w a a         

0 

### 1 1 , 2 2 u 

### n I       



## ( ,) ( ) a  ( ) w a e w a  1 2 a w b  

3 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 1 2 6 3 w a a         

## ( ,) ( ) a  ( ) w a e w a  1 2 a w b   2 

### 1 0 1 1 u t n 

c.1.7 #NUM! d.1.7

Result1: #NUM! Result2:
1- Calculate #NUM!

2- Calculate

2.1. Calculate #DIV/0!

2.2. Calculate #NUM!

2.3. Calculate #NUM!

Result: #DIV/0!

3-Calculate

3.1. Calculate #DIV/0!

3.2. Calculate #DIV/0!



### 2 6 3    

2

### 3 1 2 6 w a  

    

0 0 

### 1 11 ( ) , 2 2 2 u 

### n P t I 

      

3 0 0 

### 2 1 1 1 1 ( ) ,1 ,1

2 2 6 2 3 2

u u 

### n n n n P t I I

n n  

###  

         

        2 2 01 2 2 0 

### 2 1 1 1 1 ( ) 6 2 1 1 n



### n t n P t n t

n  

###  



###      

     

  

   

## 2 1 6 2 nn 

2

0 

## 2 1 1 1 n t n 

 1 2 2 0 

## 1 1 n 

## t n  

##     

  

1 3 2 nn  

## 2 1 6 2 nn Result #DIV/0!

4-Calculate
4.1. Calculate #NUM!

4.2. Calculate #DIV/0!

4.3. Calculate #DIV/0!

Result: #NUM!
5- Calculate

#NUM! 5.1. Calculate #DIV/0! 5.2. Calculate

5.3. Calculate #DIV/0!

5.4. Calculate #DIV/0!

Result #NUM! 

#      
4 0 0 0 

# 1 2 4 1 1 11 11 31 ( ) , , , 24 2 2 12 2 2 24 2 2 u u u



# n n n n n n n n P t I I I n n  

#    
                        

0 

## 1 11

, 24 2 2 u 

## n n I         

#    0 

## 1 2 1 1

, 12 2 2

u 

## n n n

I n   

      

#    0 

### 4 1 3 1

, 24 2 2

u 

### n n n

I n   

      

#    

#    

#    

#   2 3/2 0 0

2 2

0 0



### 1 2 5 8 1 (2 5) 1 1 1 1 ( ) , ,

72 2 2 24 2 2 1 2 5 12 12 5 12 3 1 5 1 , , 24 2 2 72 2 2 u u

u u



### n n n n n n n P t I I n n n n n n n n n I I n n  

###   

   

      

                   

         

#   0 

## 1 (2 5) 1 1

, 72 2 2

u 

## n n n

I  


      

#  

#  

2 0 

## 1 2 5 8 1 1 , 24 2 2 u 

## n n n n

I n    

      

#  

#  

2 0 

## 1 2 5 12 3 1 , 24 2 2 u 

## n n n n

I n           

#  

#  

2 0 

## 1 2 5 12 5 1 , 72 2 2 u 

## n n n n

I n          Corrected probability is:: 3 4 3/2 2 0 3 

### ( ) ( ) ( ) ( ) P t SkewnessP t KurtosisP t P t    

###       #NUM! u0= #NUM!

0.5 a= 1.5 0.5 b= 0.5 #NUM! w= #NUM!

#NUM! e.1.1. #NUM! 1.772453851 e.1.2. 0.886226925 #NUM! e.1.3 #NUM!
#NUM! e.1.4. #NUM! 0.0625 e.1.5.

#NUM! e.1.6 Calculations e.1.) Calculate 

##  3 2 2 2

2 3 

### ( ,) 1 1 5 3 1 3 11 1 , ( ) ( ) 2 2 3 2 326 3 2 3 1 1 2 6 2 w a 

### a w e w a w a I ab a a w a a

a a b b

w a w  

###    



           

                        

0 1 1 , 2 2 u n I       



## ( ,) ( ) a  ( ) w a e w a 1 2 a w b  

3 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 1 2 6 3 w a a         

## ( ,) ( ) a  ( ) w a e w a  1 2 a w b  

3 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 1 2 6 3 w a a        #NUM! e.1.7

#NUM! Result3: 

### 2 6 3    

2

### 3 1 2 6 w a  

     

### 2 6 3    

2

### 3 1 2 6 w a  

    0 0 0 

# 1 2 4 1 1 11 11 31 ( ) , , , 24 2 2 12 2 2 24 2 2 u u u 

# n n n n P t I I I                      #NUM!a= 1.5 b= 0.5
w= #NUM!

f.1.1. #NUM! f.1.2. 0.886226925 f.1.3 #NUM!
f.1.4. #NUM! -0.14583 f.1.5. -0.14583

#NUM! f.1.6 #NUM!

Calculations f.1.) Calculate 0 3 1 , 2 2 u n I        0 5 1 , 2 2 u n I       



## ( ,) ( ) a  ( ) w a e w a  1 2 a w b  

3 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 1 2 6 3 w a a        #NUM! f.1.7 #NUM!

#NUM! Result4: #NUM! 

### 2 6 3    

2

### 3 1 2 6 w a  

    a= -0.5 a= 0.5

b= 1 b= 1
w= #NUM! w= #NUM!
g.1.1. #NUM! h.1.1. g.1.2. #NUM! h.1.2. g.1.3 #NUM! h.1.3 g.1.4. #NUM! h.1.4. g.1.5. -1.5625 h.1.5.
g.1.6 #NUM! h.1.6 g.1.) Calculate h.1.) Calculate

Calculations Calculations 0 1,1 2 u n I       



## ( ,) ( ) a  ( ) w a e w a  1 2 a w b  

3 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 1 2 6 3 w a a         

## ( ,) ( ) a  ( ) w a e w a  1 2 a w b  

3 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 1 2 6 3 w a a        g.1.7 #NUM! h.1.7

Result5: #NUM! Result6:



### 2 6 3    

2

### 3 1 2 6 w a  

     

### 2 6 3    

2

### 3 1 2 6 w a  

    #NUM!

1.772454 #NUM!
#NUM! 0.0625

#NUM! 0 1,1 2 u n I       

5 3 1 2 3 2 3 a a   

2 

### 3 11 1 2 6 3 w a a        #NUM!

#NUM! 

### 2 6 3     3 1 2 6 w a  

    